{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":45976,"title":"Evaluate the Struve function","description":"The \u003chttps://en.wikipedia.org/wiki/Struve_function Struve function\u003e *H*_a(x) is a solution to an inhomogeneous form of Bessel's equation:\r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/246a7bab900d24f188a7edcec59042852728d747\u003e\u003e\r\n\r\nThe Struve function appears in applications of optics and loudspeaker design. I encountered it in developing a model of turbulence in a strongly stratified fluid (e.g., a lake with a large temperature difference). \r\n\r\nEvaluate the Struve function *H*_p(x) for given values of the order p and argument x. 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src=\"data:image/png;base64,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\" alt=\"H_p(x)\" style=\"width: 40px; height: 20px;\" width=\"40\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 190.383px 7.79167px; transform-origin: 190.383px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a solution to an inhomogeneous form of Bessel's equation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38.95px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.475px; text-align: left; transform-origin: 384px 19.475px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; 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alt=\"Differential equation from the Wikipedia link\" style=\"width: 271.5px; height: 39px;\" width=\"271.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 370.333px 7.79167px; transform-origin: 370.333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Struve function appears in applications of optics and loudspeaker design. I encountered it in developing a model of turbulence in a strongly stratified fluid (e.g., a lake with a large temperature difference).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9083px; text-align: left; transform-origin: 384px 10.9083px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.5167px 7.79167px; transform-origin: 87.5167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEvaluate the Struve function\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"H_p(x)\" style=\"width: 40px; height: 20px;\" width=\"40\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90.625px 7.79167px; transform-origin: 90.625px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for given values of the order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.0667px 7.79167px; transform-origin: 47.0667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and argument \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function Hp = StruveFn(p,x)\r\n  Hp = f(x,p);\r\nend","test_suite":"%%\r\nx = 0;\r\np = randi(8,1);\r\nHp_correct = 0;\r\nassert(isequal(StruveFn(p,x),Hp_correct))\r\n\r\n%%\r\nx = 0.4;\r\np = 0;\r\nHp_correct = 0.2501497138634162;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = 10;\r\np = 0;\r\nHp_correct = 0.1187436836875042;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = rand(1);\r\np = 1/2;\r\nHp_correct = sqrt(2/(pi*x))*(1-cos(x));\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = 4.2;\r\np = 1;\r\nHp_correct = 1.036818631956923;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = pi^2;\r\np = 3;\r\nHp_correct = 4.10841348624688;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%%\r\nx = rand(1);\r\np = 1;\r\nHp_approx = 2/pi - besselj(0,x) + (16/pi-5)*sin(x)/x + (12-36/pi)*(1-cos(x))/x^2;\r\nassert(abs(StruveFn(p,x)-Hp_approx)/Hp_approx\u003c0.002)","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":"2021-01-02T18:43:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-21T16:48:35.000Z","updated_at":"2026-01-09T13:52:55.000Z","published_at":"2020-06-21T17:11:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Struve_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eStruve function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"H_p(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\bf H}_p(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a solution to an inhomogeneous form of Bessel's equation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Differential equation from the Wikipedia link\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^2\\\\frac{d^2y}{dx^2}+x\\\\frac{dy}{dx}+(x^2-p^2)y=\\\\frac{4(x/2)^{p+1}}{\\\\sqrt{\\\\pi}\\\\Gamma(p+1/2)}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Struve function appears in applications of optics and loudspeaker design. I encountered it in developing a model of turbulence in a strongly stratified fluid (e.g., a lake with a large temperature difference).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEvaluate the Struve function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"H_p(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\bf H}_p(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for given values of the order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and argument \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51625,"title":"Solve an ODE: diffusion problem 2","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQQAAAAmCAYAAAAiG+WDAAAGfUlEQVR4nO2dbXHrOhCGXw5mEAIhEARBEAZhEAaduQjy5xIIhHDIHAbGUAq9P+T3aqsjy7K1Uq10n5nMtLHrj931fklWAcMwDMMwDMMwDMMwDMMwjB/iAOCUud9hYZ8jgKH4ivriA+6+Sxng9PDb5Ac4+eXIMGZfZwBX9Sv6hRwBjACeAO4APjEv2COALwCvxPGu0z4fite4d+4ALoXHGOB0MAK4wcnwXnjMXrjA2d0dXgZzwSllXze8l8xOcPd0g5NR9SBxghPuY/qdwv6cOfkHlg31Ne1T+oD0wh1OYSUMcDIfp5/peL+Ql7X1DJ0fg9Bj+v05s/+Sfd3RfzA6w9nCC07/Jzi5pIJ1MTTCT/gS4Iy0MsZp+1xad5i207DfnSvS2VIuTzi5naffD/AOYqk86xkGJClDBp2Yk821rxFelr1xgZdJeI+0kyoOj575sbTjBKNWan9mGJpejL0NjfpcExpnaQTnQ/FZfEXz7FWGNPBce8m1rxPms9w9w0Agg0O4nZmjui554tzUnspIXcgD+tnBWsfVCta7pTBFrln77lGGsizKzYLW2NcL/fUT7kiX7IAvmTQy0/9hWrJGGS+kDWqAfnYA7NOY6alL01Lp8Wv2XPYoQxr/mLn/WvuijfeSJfD+UiU74EuqNc/uIoxKa5WREu4Zyl5rYo/GTKWUwqxLVbkR9ihDZqi5UXytfdUKULWQQTrVI5D7Fd3bFX4Ig8oYxXdXzD/wl4WLBJxiazRyWhjzAHftUg4HuPuOCZ3DtFvgeW7w6d+X+O4GfedQQ4aUWUznR/h7GcT+/E5GuSe+D63NscW+XtAp61pAHS096CexX5E+KXSmajxgjjKAZSMdUCc9q+0Q6CAfcMLmkA9HVMIIxjR/61DjBV7m1MML3x2CNtoyPMIHlbCxKu1L1roH+Pt74ns05PepBu0W++K1bEHqqeSTi5RJrkPIzfCTSEPsYYy7lkMY4CN0qAApo9BRnmb+Zi1SsTWcgERThkc4uclyh5H7Pp1DyjZWatL4a48E8L63dOTlA7r1s2bkSJ4vFZxlM1ZlZIr9g16GZWqlu3POQJ4zZszcVupM5QNV2zFryvABZ5Sylh3gnQGhgceiGLOL2uk8ZbxFvhoZwpqgIR3CkgNTdQitlJGLTCVjHwoqTKvDzxql0xnMyeCe2K7lEDQd80/IkDJithA6GxptWHLJCFc7M9LK5logHUKqVyJHpoodgjzYXqZ3ytS55JOrdBn95zwx+wcxg9VyCDyHxshMaxkC3wNLODdAPvShcV8S27ShXGo7Hg0YIJr2EDg1uYUycsmNbuPCfjl1Iqdrp9JnKfDYQ6/hEOSYs4ZjbilD4O86NpSFLIfC7OcjsU2bnhyCDFRNRhmAtsrQQrP+zanblxpeGg6htWPW7sNI440dkzKMlVxVZtrNwGzkp3oIaxyRtInU3+Xul4WsJXtB05iXHvYc70uFlDzIqaZlDbQdwlIDLGWwLUvWEufdepRBZo0pPamOEvJAYaNnz2gaM+8/5hDDobI573tc2J5DqgNfA00ZLhluquSS21q8Hl8yo7T1KAPQ+F0GWff1tFZBC4dAZyBLKr4dGHuZhhOZtrJ22m4pmjKUKWvMjijDWHRsNVWbPNFXNrzmbcfi7KC1MrTQNGbZyaXAT/BDZ+Fkm7mVe+7YHt1/wjFrylA+8LEoxggWO9fad2hKYCazl9G0XFLrIVB+KvfEdKRVmqqFpjHzPXlZ543wD6Z0CLHuOWGU3DIDbsubpqVoypDDpbGGoSwnYs5ubjp4DaijnoIf4ZJyT7j74IpJqiMm9Ny9eUzthtgB/kWvMC0bpm0569eN2GbYjLAtU1lNGaYmMaWcnXQWLUZWHtjP5LstyBfuaKtqDWipjL2tmrPEXlf7OWPbLMPUlOlatJJhKgtlxK65MhRhrb03m9kNsi4x9HhgXZZAQ5XrWL4Tqf4BnUWLDPXZ6DxdwBdNZARi/dHD2409wdGJuRT4BmeYzCLYo+hh5twWmIX+i7/XL+CCvrXnXWgtfPs2hJNe2NXuae5BTxzgDDBMT+WYO+s/rrPwjsj7/YPvGSkdYe1RFb6a3css3CZw8Qp2z7c2v4x8+E9WjsF38n8IcEz8XY1VOoR/4EcirnA2WLtncoRfi8EIYCf9ivesVfdKaIwD/Iy339Dgki9JcRm1Jv9xCGbnhmEYhmEYhmEYhmEYhmEY6/kPn1M7vflLTCQAAAAASUVORK5CYII=\" alt=\"f''+a(f+eta f') = 0\" style=\"width: 130px; height: 19px;\" width=\"130\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral(f,{eta,-inf,inf}) = 1\" style=\"width: 78px; height: 44px;\" width=\"78\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion2ODE(eta,a)\r\n%  eta = independent variable\r\n%  a   = positive constant\r\n  f = a*sqrt(eta);\r\nend","test_suite":"%%\r\na = 1/2; \r\neta = 0:0.2:1;\r\nf_correct = [0.282094791773878 0.279287901697234 0.271033696776216 0.257815227404741 0.240385324709827 0.219695644733861];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2; \r\neta = 0.5;\r\nf_correct = 0.439391289467722;\r\nassert(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13)\r\n\r\n%%\r\na = 1;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.396080211793656 0.388528585315836 0.359548380672770 0.292234407541508 0.239309153606614];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2;\r\neta = 1:4;\r\nf_correct = [0.207553748710297 0.010333492677046 0.000069626525973 0.000000063491173];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 3/4;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [0.345377564870647 0.334028296534584 0.011822159682599];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 1;\r\neta = rand/120;\r\nf_correct = polyval([1/10321920 0 -1/645120 1/46080 0 -1/3840 0 1/384 0 -1/48 0 1/8 0 -1/2 0 1]./(sqrt(2*pi)),eta);\r\nf = diffusion2ODE(eta,a);\r\nassert(all(abs(f-f_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-01T14:31:00.000Z","updated_at":"2021-05-01T14:34:52.000Z","published_at":"2021-05-01T14:34:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a(f+eta f') = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime+a(f+\\\\eta f\\\\prime)=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral(f,{eta,-inf,inf}) = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty fd\\\\eta=1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51745,"title":"Solve an ODE: equation A","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith  and . The function should return the values of  at the specified values of . One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.3px; transform-origin: 407px 59.3px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 8px; transform-origin: 209.417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y\u0026quot;-xy = 0\" style=\"width: 74px; height: 36.5px;\" width=\"74\" height=\"36.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 8px; transform-origin: 14.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 8px; transform-origin: 128.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 8px; transform-origin: 80.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2333px 8px; transform-origin: 62.2333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEA(x,x0,y0,x1,y1)\r\n  y(x0) = y0; \r\n  y(x1) = y1; \r\n  y = f(x,x0,y0,x1,y1);\r\nend","test_suite":"%%\r\nx0 = -3; \r\ny0 = 1;\r\nx1 = 3;\r\ny1 = 0.2;\r\nx  = linspace(x0,x1,7);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [1 -0.608544735138462 -1.416513846161609 -0.930561687980622 -0.339539877942166 -0.041385541117497 0.2];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -5; \r\ny0 = 2;\r\nx1 = 5;\r\ny1 = 0;\r\nx  = linspace(x0,x1,11);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [2 -0.400646543833150 -2.159956361501116 1.296651996575315 3.053707895612597 2.024329503005815 0.771421025528832 0.199130370987114 0.037568752980204 0.005346964905914 0];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -4; \r\ny0 = -1;\r\nx1 = 2;\r\ny1 = 0.3;\r\nx  = linspace(x0,x1,6);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [-1 -4.088820829832713 5.996215644909088 6.297137060461841 2.304927532867409 0.3];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -7; \r\ny0 = 0;\r\nx1 = 3;\r\ny1 = 0.1;\r\nx  = linspace(x0,x1,6);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [0 -0.004972738967217 0.002891447704152 -0.005345046731767 0.007070410943182 0.1];\r\nassert(all(abs(y-y_correct) \u003c 1e-14))\r\n\r\n%% anti-cheating--product of two values\r\nx0 = -2; \r\ny0 = 1;\r\nx1 = 2;\r\ny1 = 0.1;\r\nx  = [-1 1];\r\nz  = prod(solveODEA(x,x0,y0,x1,y1));  \r\nz_correct = 1.336786968358133;\r\nassert(all(abs(z-z_correct) \u003c 1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-15T13:16:15.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":"2021-05-14T12:28:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-11T04:16:02.000Z","updated_at":"2025-09-02T13:25:13.000Z","published_at":"2021-05-11T04:19:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot;-xy = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} – x y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54670,"title":"Solve an ODE: precocious pair’s porcine pursuit","description":"In our previous encounters with Matilda and Labrun, the scintillating siblings collected candy wrappers, amused others with card tricks, and found interesting relations involving house numbers on their street. \r\nBut now their pet pig has run away, and the pair must catch her! They start a distance  away from the pig, which runs at speed  in a direction perpendicular to the line connecting the initial positions of the pig and the siblings. Matilda and Labrun run at speed , always in a direction pointing at the current position of the pig. \r\nWrite a function that takes the distance  and the two speeds and returns the time required for Matilda and Labrun to catch their pet. Return Inf if the pair will not catch the pig, and please ignore the impracticality of reporting times to the nearest microsecond. \r\nThis problem is adapted from a problem in Advanced Mathematical Methods for Scientists and Engineers by Bender and Orzsag.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 237.45px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.725px; transform-origin: 407px 118.725px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.133px 7.79167px; transform-origin: 236.133px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn our previous encounters with Matilda and Labrun, the scintillating siblings \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/53004\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ecollected candy wrappers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/51451\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eamused others with card tricks\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.79167px; transform-origin: 17.5px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/51251\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003efound interesting relations involving house numbers on their street\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.292px 7.79167px; transform-origin: 266.292px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBut now their pet pig has run away, and the pair must catch her! They start a distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ed\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104.233px 7.79167px; transform-origin: 104.233px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e away from the pig, which runs at speed \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 336.5px 7.79167px; transform-origin: 336.5px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a direction perpendicular to the line connecting the initial positions of the pig and the siblings. Matilda and Labrun run at speed \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 197.208px 7.79167px; transform-origin: 197.208px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, always in a direction pointing at the current position of the pig. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.725px; text-align: left; transform-origin: 384px 31.725px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.383px 7.79167px; transform-origin: 122.383px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ed\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.775px 7.79167px; transform-origin: 254.775px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the two speeds and returns the time required for Matilda and Labrun to catch their pet. Return \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.79167px; transform-origin: 11.55px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 11.55px 8.25px; transform-origin: 11.55px 8.25px; \"\u003eInf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 311.95px 7.79167px; transform-origin: 311.95px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e if the pair will not catch the pig, and please ignore the impracticality of reporting times to the nearest microsecond. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 133.033px 7.79167px; transform-origin: 133.033px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem is adapted from a problem in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.492px 7.79167px; transform-origin: 194.492px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eAdvanced Mathematical Methods for Scientists and Engineers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.79167px; transform-origin: 47.4583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eby Bender and Orzsag.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function t = pigPursuit(d,V,v)\r\n%  d = initial distance between pig and M\u0026L. The pig runs perpendicular to the line connecting the initial positions\r\n%  V = pig speed\r\n%  v = M\u0026L speed\r\n%  t = time till capture\r\n\r\n  y = hypot(d/V,d/v);\r\nend","test_suite":"%%\r\nd = 5;                      %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 4.1;                    %  M\u0026L speed (m/s)\r\nt_correct = 25.308642;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 5;                      %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 4.5;                    %  M\u0026L speed (m/s)\r\nt_correct = 5.294118;       %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 5;                      %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.1;                    %  M\u0026L speed (m/s)\r\nt_correct = 25.247525;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 5;                      %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.2;                    %  M\u0026L speed (m/s)\r\nt_correct = 12.745098;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 5;                      %  M\u0026L speed (m/s)\r\nt_correct = 5.555556;       %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 4.3;                    %  M\u0026L speed (m/s)\r\nt_correct = 17.269076;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.5;                    %  M\u0026L speed (m/s)\r\nt_correct = 10.476190;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 6;                      %  M\u0026L speed (m/s)\r\nt_correct = 5.454545;       %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 5;                      %  M\u0026L speed (m/s)\r\nt_correct = 11.111111;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 6;                      %  M\u0026L speed (m/s)\r\nt_correct = 6;              %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.01;                   %  M\u0026L speed (m/s)\r\nt_correct = 1000.999001;    %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.001;                  %  M\u0026L speed (m/s)\r\nt_correct = 10000.9999;     %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 100*rand;\r\nV = 6*rand;\r\nv = V;\r\nassert(isinf(pigPursuit(d,V,v)))\r\n\r\n%%\r\nd = 100*rand;\r\nV = 6*rand;\r\nv = V*rand;\r\nassert(isinf(pigPursuit(d,V,v)))\r\n\r\n%%\r\nfiletext = fileread('pigPursuit.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'import'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-24T14:38:30.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2022-05-24T14:38:30.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-24T14:03:33.000Z","updated_at":"2022-05-24T14:38:30.000Z","published_at":"2022-05-24T14:05:54.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn our previous encounters with Matilda and Labrun, the scintillating siblings \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/53004\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ecollected candy wrappers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/51451\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eamused others with card tricks\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/51251\\\"\u003e\u003cw:r\u003e\u003cw:t\u003efound interesting relations involving house numbers on their street\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBut now their pet pig has run away, and the pair must catch her! They start a distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"d\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ed\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e away from the pig, which runs at speed \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in a direction perpendicular to the line connecting the initial positions of the pig and the siblings. Matilda and Labrun run at speed \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, always in a direction pointing at the current position of the pig. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"d\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ed\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the two speeds and returns the time required for Matilda and Labrun to catch their pet. Return \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if the pair will not catch the pig, and please ignore the impracticality of reporting times to the nearest microsecond. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem is adapted from a problem in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eAdvanced Mathematical Methods for Scientists and Engineers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eby Bender and Orzsag.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51815,"title":"Solve an ODE: equation C","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith  and . The parameter  is a constant. The function should return the values of  at the specified values of .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.2917px; transform-origin: 407px 59.2917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 7.91667px; transform-origin: 209.417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\" style=\"width: 179.5px; height: 37.5px;\" width=\"179.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.725px 7.91667px; transform-origin: 51.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.967px 7.91667px; transform-origin: 169.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.95px 7.91667px; transform-origin: 50.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEC(x,a,Xbc,bc)\r\n%  y   = values of the function\r\n%  x   = independent variable\r\n%  a   = parameter in the equation\r\n%  Xbc = [x0 x1], values of x where the boundary conditions are specified\r\n%  bc  = [y0 y1], values of the function at x0 and x1, respectively\r\n\r\n   y = f(x,a,Xbc,bc);\r\nend","test_suite":"%% \r\nx   = [-1/3 -1/4 0 1/4 1/3];\r\na   = 1;\r\nXbc = [-1/2 1/2];\r\nbc  = [2 4];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [2.599319657044238 2.854101966249684 3.464101615137754 3.854101966249684 3.932652990377571];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx   = [-0.6 -0.2 -0.05 0.12 0.2];\r\na   = sqrt(3);\r\nXbc = [-0.7 0.3];\r\nbc  = [0 1];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [0.237055225759061 0.877546669526703 0.995431932094838 1.046546024897035 1.039090864471318];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx   = [-0.8 -0.6 -0.3 -0.15 0.1 0.25 0.375];\r\na   = 2;\r\nXbc = [-0.8 0.4];\r\nbc  = [-1 3];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [-1 1.68647951290722 4.138417234449975 4.687455882945745 4.630189304757032 4.024530246664777 3.199461161409147];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx   = [-0.7 -0.45 -0.2 0.05 0.3 0.55 0.8];\r\na   = pi;\r\nXbc = [-0.9 0.9];\r\nbc  = [-1 1];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [1.764854605944604 2.706629373469937 1.6090059417224112 -0.4259046519843118 -2.225041323571773 -2.630850806938173 -0.6162122684969365];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\na   = 5;\r\nXbc = [-1/2 1/2]; \r\nbc  = [-1 1];\r\ny   = 1;\r\nx   = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na   = 5;\r\nXbc = [-1/2 1/2]; \r\nbc  = [-1 1];\r\ny   = 1;\r\nx   = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na   = 2*randi(4)+1;\r\nXbc = [-3/4 3/4]; \r\nbc  = [-2 2];\r\nx   = rand-3/4;\r\nym  = solveODEC(-x,a,Xbc,bc);\r\nyp  = solveODEC(x,a,Xbc,bc);\r\nassert(abs(ym+yp)\u003c1e-13)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-25T04:18:45.000Z","updated_at":"2021-05-25T04:23:56.000Z","published_at":"2021-05-25T04:23:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(1-x^2)\\\\frac{d^2y}{dx^2} -x \\\\frac{dy}{dx}+a^2 y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":231,"title":"Differential equations I","description":"Given a function handle |f| an initial condition |y0| and a final time |tf|, solve numerically the differential equation\r\n\r\n  dy/dt = f(y)\r\n\r\nfor the function |y(t)| between |t=0| and |t=tf|. Give as a result |res=y(tf)|.\r\n\r\nExample:\r\n\r\n   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\r\n\r\n =\u003e y(tf) = 1/e = 0.367879441171442\r\n\r\nRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].","description_html":"\u003cp\u003eGiven a function handle \u003ctt\u003ef\u003c/tt\u003e an initial condition \u003ctt\u003ey0\u003c/tt\u003e and a final time \u003ctt\u003etf\u003c/tt\u003e, solve numerically the differential equation\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003edy/dt = f(y)\r\n\u003c/pre\u003e\u003cp\u003efor the function \u003ctt\u003ey(t)\u003c/tt\u003e between \u003ctt\u003et=0\u003c/tt\u003e and \u003ctt\u003et=tf\u003c/tt\u003e. Give as a result \u003ctt\u003eres=y(tf)\u003c/tt\u003e.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\u003c/pre\u003e\u003cpre\u003e =\u003e y(tf) = 1/e = 0.367879441171442\u003c/pre\u003e\u003cp\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/p\u003e","function_template":"function res = deqnsolve(f,y0,tf)\r\n  res = 0;\r\nend","test_suite":"%% \r\nf = @(x) -x;\r\ntf =1;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(-1)) \u003c 1e-5)\r\n\r\n%% \r\nf = @sin;\r\ntf =1;\r\ny0 =1/2;\r\nassert(abs(deqnsolve(f,y0,tf)-2*acot(exp(-1)*cot(1/4))) \u003c 1e-5)\r\n\r\n%% \r\nf = @(x) 1/(x+1);\r\ntf =6;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-3) \u003c 1e-5)\r\n\r\n%% a randomized one\r\na = rand*0.9;\r\nf = @(x) x-a*x^2;\r\ntf = rand+1.5;\r\ny0=1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(tf)/(1-a+a*exp(tf))) \u003c 1e-5)","published":true,"deleted":false,"likes_count":3,"comments_count":5,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":190,"test_suite_updated_at":"2012-02-02T15:10:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-02T14:57:14.000Z","updated_at":"2026-03-13T19:43:52.000Z","published_at":"2012-02-02T15:20:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a function handle\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e an initial condition\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and a final time\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, solve numerically the differential equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[dy/dt = f(y)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey(t)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=tf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Give as a result\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eres=y(tf)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   f = @(x) -x;\\n   tf= 1;\\n   y0= 1;\\n\\n =\u003e y(tf) = 1/e = 0.367879441171442]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":50609,"title":"Solve an ODE: equidimensional equation","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 140.45px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 70.225px; transform-origin: 407px 70.225px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.375px 7.79167px; transform-origin: 152.375px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSolve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.1833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5917px; text-align: left; transform-origin: 384px 18.5917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x^2 y''(x) + a x y'(x) + b y(x) = 0\" style=\"width: 144px; height: 37px;\" width=\"144\" height=\"37\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64.2667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32.1333px; text-align: left; transform-origin: 384px 32.1333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.79167px; transform-origin: 14.3917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x_0) = y_0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dy/dx = y'_0\" style=\"width: 74.5px; height: 20px;\" width=\"74.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.79167px; transform-origin: 9.71667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x = x0\" style=\"width: 40.5px; height: 20px;\" width=\"40.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 55.225px 7.79167px; transform-origin: 55.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The parameters \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.242px 7.79167px; transform-origin: 160.242px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are constants, and the value of the function and its derivative at the point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.79167px; transform-origin: 7.7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 7.7px 8.25px; transform-origin: 7.7px 8.25px; \"\u003ex0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.9px 7.79167px; transform-origin: 52.9px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are specified as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.79167px; transform-origin: 7.7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 7.7px 8.25px; transform-origin: 7.7px 8.25px; \"\u003ey0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.79167px; transform-origin: 11.55px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 11.55px 8.25px; transform-origin: 11.55px 8.25px; \"\u003eyp0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 205.767px 7.79167px; transform-origin: 205.767px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, respectively. Your function should return the value of the function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.00833px 7.79167px; transform-origin: 9.00833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = equidimODE(x,a,b,y0,yp0,x0)\r\n%  a,b = parameters in the ODE\r\n%  x   = point at which the solution y is to be evaluated\r\n%  x0  = point at which the conditions are specified\r\n%  y0  = value of the solution at x = x0\r\n%  yp0 = value of the derivative at x = x0\r\n\r\n y = f(x,a,b,y0,yp0);\r\nend","test_suite":"%%\r\na  = 2; b  = -1; x   = 4;\r\nx0 = 1; y0 = 1;  yp0 = 0;\r\ny_correct = 1.733830915729880;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = 3; b  = -1; x   = 4;\r\nx0 = 2; y0 = 1;  yp0 = 3;\r\ny_correct = 3.593733292875542;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = 1; b  = 1; x   = 4;\r\nx0 = 1; y0 = 1; yp0 = 0;\r\ny_correct = 0.183456974743302;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = 0.5; b  = 1; x   = 6;\r\nx0 = 0.2; y0 = 1; yp0 = -1;\r\ny_correct = -2.149237864206678;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = -3; b  = 4; x   = 5;\r\nx0 = 1;  y0 = 0; yp0 = 1;\r\ny_correct = 40.235947810852508;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = rand; b  = 0; x   = 5;\r\nx0 = 1;    y0 = 1; yp0 = 1;\r\ny_correct = y0+(x0*yp0/(1-a))*((x/x0)^(1-a)-1);\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-02-27T23:20:45.000Z","updated_at":"2024-12-09T20:16:25.000Z","published_at":"2021-02-28T00:00:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSolve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x^2 y''(x) + a x y'(x) + b y(x) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^2 {d^2y\\\\over dx^2} + a x {dy\\\\over dx} + b y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x_0) = y_0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dy/dx = y'_0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003edy/dx = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x = x0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = x_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The parameters \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are constants, and the value of the function and its derivative at the point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e are specified as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eyp0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e, respectively. Your function should return the value of the function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":50928,"title":"Solve an ODE: separable equation","description":"Solve the following ordinary differential equation:\r\n\r\nwith the initial condition .The test suite will ask for the value of the solution  at point . Functions such as ode45, ode23, and ode15S are not allowed. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 59px; transform-origin: 407.5px 59px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150.433px 7.66667px; transform-origin: 150.433px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSolve the following ordinary differential equation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 18px; text-align: left; transform-origin: 384.5px 18px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dy/dx = f(x)/g(y)\" style=\"width: 65px; height: 36px;\" width=\"65\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21.5px; text-align: left; transform-origin: 384.5px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.9167px 7.66667px; transform-origin: 73.9167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith the initial condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 156.35px 7.66667px; transform-origin: 156.35px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.The test suite will ask for the value of the solution \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.8333px 7.66667px; transform-origin: 26.8333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.675px 7.66667px; transform-origin: 84.675px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Functions such as ode45, ode23, and ode15S are not allowed. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = separableODE(x,f,g,x0,y0)\r\n%  x  = point at which the value of the solution is requested\r\n%  f  = function of x\r\n%  g  = function of y\r\n%  x0 = point at which the initial condition is specified\r\n%  y0 = value of the solution at x = x0\r\n\r\ny = y0+(x-x0)*f(x0)/g(y0);\r\nend","test_suite":"%%\r\nf = @(x) x;\r\ng = @(y) y;\r\nx0 = 0; y0 = 4; x = 4; \r\ny_correct = sqrt(32);\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) cos(x);\r\ng = @(y) exp(y);\r\nx0 = 0; y0 = 1; x = 7*pi; \r\ny_correct = 1;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) ones(size(x));\r\ng = @(y) 1./y;\r\nx0 = 2; y0 = 3; x = 2.5; \r\ny_correct = 4.946163812100385;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) sinh(x);\r\ng = @(y) cosh(y);\r\nx0 = 0; y0 = 0.881373587019543; x = 5; \r\ny_correct = 5.000090791616095;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) exp(-x.^2);\r\ng = @(y) sqrt(y);\r\nx0 = 0; y0 = 4; x = 3; \r\ny_correct = 4.431659465773041;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) 1./(1+x.^2);\r\ng = @(y) log(y)+1;\r\nx0 = 0; y0 = 1; x = 1; \r\ny_correct = 1.622607687386726;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) tan(x);\r\ng = @(y) sec(y);\r\nx0 = asec(-1-sqrt(2)); y0 = 3*pi/4; x = 2.302554350306210; \r\ny_correct = 7*pi/8;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\t\r\n%%\r\nfiletext = fileread('separableODE.m');\r\nnoODEfns  = ~contains(filetext, 'ode45') \u0026\u0026 ~contains(filetext, 'ode7') \u0026\u0026 ~contains(filetext, 'ode8') \u0026\u0026 ~contains(filetext, 'ode2') \u0026\u0026 ~contains(filetext, 'ode1');\r\nassert(noODEfns, 'No built-in ODE solvers allowed')\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":"2022-01-17T04:12:35.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-13T17:05:52.000Z","updated_at":"2022-01-17T04:12:35.000Z","published_at":"2021-03-13T17:09:14.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSolve the following ordinary differential equation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dy/dx = f(x)/g(y)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{dy}{dx}=\\\\frac{f(x)}{g(y)}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith the initial condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.The test suite will ask for the value of the solution \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Functions such as ode45, ode23, and ode15S are not allowed. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51387,"title":"Solve an ODE: second-order linear equation with constant coefficients","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 190.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 95.2917px; transform-origin: 407px 95.2917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 314.317px 7.91667px; transform-origin: 314.317px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve a second-order linear ordinary differential equation with constant coefficients \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.91667px; transform-origin: 17.5px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"ay\u0026quot;+by'+cy = 0\" style=\"width: 131.5px; height: 37.5px;\" width=\"131.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.625px; text-align: left; transform-origin: 384px 31.625px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.458px 7.91667px; transform-origin: 110.458px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith two of the three conditions: (1) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3833px 7.91667px; transform-origin: 14.3833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, (2) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'(x0) = y'0\" style=\"width: 74px; height: 20px;\" width=\"74\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and (3) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 126.783px 7.91667px; transform-origin: 126.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Two of the three elements of the vector bc will be assigned numerical values, and the third will be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.95px 7.91667px; transform-origin: 50.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 374.975px 7.91667px; transform-origin: 374.975px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEquations of this form appear in many applications, such as spring-mass-damper systems, RLC circuits, small-amplitude oscillations of a pendulum, and steady advection, dispersion, and decay of a contaminant in a river or groundwater. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc)\r\n%  x     = values of the independent variable at which the dependent variable y is requested\r\n%  coeff = coefficients in the ODE [a b c]\r\n%  bc    = boundary conditions [y0 yp0 y1] = [y(x0) y'(x0) y(x1)]\r\n%  Xbc   = values of x at which the boundary conditions are specified [x0 x1]\r\n\r\n   y = f(x,coeff,bc,Xbc);\r\nend","test_suite":"%%\r\nx = (0:0.25:2)*pi;\r\ncoeff = [1 0 1];\r\nbc = [0 1 NaN];\r\nXbc = [0 NaN];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\na = 1/sqrt(2);\r\ny_correct = [0 a 1 a 0 -a -1 -a 0]; \r\nassert(all(abs(y-y_correct)\u003c1e-15))\r\n\r\n%%\r\nx = [0 0.1 2.3 4.56];\r\ncoeff = [2 7 -15];\r\nbc = [3 0 NaN];\r\nXbc = [0 NaN];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [3 3.101061786097092 72.69322003332921 2156.513387836726]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [1/7 1/2 5/6];\r\ncoeff = [9 24 16];\r\nbc = [1 NaN 0];\r\nXbc = [0 1];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [0.7084846608207754 0.256708559516296 0.05486549796798426]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [1.2 3.4 4.7 5];\r\ncoeff = [1 2 3];\r\nbc = [NaN 2 3];\r\nXbc = [1 5];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [394.630646389682 -43.24486540256236 -1.2288043607505912 3]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [0.001 0.01 0.015 0.1 0.5 0.7512];\r\ncoeff = [1 100 1];\r\nbc = [2 NaN 1];\r\nXbc = [0 1];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [1.9057927709196938 1.374168410483207 1.2308202441263283 1.0090865186701083 1.005013023466313 1.002491347110156]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [-1.42 -0.56 0 1.8 2.78 4];\r\ncoeff = [9 12 4];\r\nbc = [NaN pi exp(1)];\r\nXbc = [-2 4];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [25.647244651987194 21.17888752808947 (7*exp(5)-12*pi)/(15*exp(4/3)) 8.21698074712973 5.101899094149195 exp(1)]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nA = 3; x1 = 5; b = 2;\r\nx = x1*rand(1,4);\r\ncoeff = [1 0 -b^2];\r\nbc = [A NaN 0];\r\nXbc = [0 x1];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = A*sinh(b*(x1-x))/sinh(b*x1); \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))","published":true,"deleted":false,"likes_count":3,"comments_count":1,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-10T17:15:55.000Z","updated_at":"2021-04-10T17:23:45.000Z","published_at":"2021-04-10T17:23:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve a second-order linear ordinary differential equation with constant coefficients \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"c\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ay\u0026quot;+by'+cy = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea \\\\frac{d^2y}{dx^2}+b\\\\frac{dy}{dx}+cy=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith two of the three conditions: (1) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, (2) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(x0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(x_0) = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and (3) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Two of the three elements of the vector bc will be assigned numerical values, and the third will be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEquations of this form appear in many applications, such as spring-mass-damper systems, RLC circuits, small-amplitude oscillations of a pendulum, and steady advection, dispersion, and decay of a contaminant in a river or groundwater. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51087,"title":"Solve an ODE: equation for a 2D laminar jet","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f’’’ + 2(f’2+ff”) = 0\" style=\"width: 147.5px; height: 20px;\" width=\"147.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f''(0) = 0\" style=\"width: 115px; height: 19px;\" width=\"115\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIoAAAAmCAYAAAAbWQPxAAAD3klEQVR4nO2aa7GrMBSFlwccYKAGquAowAEOcFAL1VAJeMACGmqB+wPWsMvNYwcSCDP5Zpg5PU0hj521HwQoFAqFQqFQKBQKqAE8le3qgPs+ALx29SgeHXRjKzh4ABgB9ADeAL4AWkfbCcAQcO8eQHWwj0epAHwANBf3IxYVgD/MG6Bb/k7KE/PCf5bP7fL5C/Pivpbv34p70wCvNhJSYTbw5JOamA7z+rwxr98f5nElG1u1PPCL1ZX8YTaE3vKbcfn+obj3iPwWpcbveO/GG+aNyrWckGDOO/yqiQ+6HU37N+zGdjUd8u2bC27iCWaVbpbvoqs4LVDrt+mWfGpSL+1yDR4rhI07F6jmNiPnuCZETB5ofRP0MjxArybaYPcq7tBHiVSTztFuwKoqUfgE3pDW6pM0tnMNJgc48T51zAXGJr4YRLbbPbYWazpFtzOK/7WwG0IDnZxRqULdTrX8ln3x/f5oengXgyY9dAbQiXa2EocXTqq0uo/4v89na1wU7x3ar8lw9TBPSoPV0OU1WNrbYO1oD12EKyRGkuPUGoqmhOFE3ix2wNljXkQtNKzX0pcWa9BGxZMTI2Mr07VtH7OvElcftJc229w+z7VhW+y7vxHGJ7ai2hFcUfkWxglbiWQVdbv4TM9ZOeaEPTbttRPE3+whhqKEuEuOzWfYT0Q0FMp2ilpCyH17uINpRvA0FiqAbUdRnbQFNSrrHQJaaSiusUnFPWQorHFEzbUFIYbiG4ysNmr8s2yvcak0lFzrPRI5D6fEKDIfT1Fe1xrK9h2TDdnfEX6loDvRRPx3MpRTsx5gfamnqYnsIdRQNHUcGX/4dgknSpNR7MnQ5HPOzHrkurkMW9vOCy0zVVWSbzF9SBfoG5DcTT4lDIk7eN89nJ31SGV1KQXHdDhRiZZjWwjJJOh3B9gHxV1vyoJsz9duggHX1lFCXf9p73qYXmqleQ/aF4fAb+GvxzpxrNIy6+HO8xkL3Zl2bKkC+lRQVWyZT4NIaiKLManOY9ClaBaLZ0Nc8ixPx/HgkUwVWe6Xh3k0cNJzOy/jw3UehYpzeEx8SLQ3ixa0b5mB9RScyUhe+H9nVPgN2OTVGdrbeCH9PKRCnjKUJ9y+iGT43I2p5ZYSGKJaVAX6bt9va6wvOTXtt1CN7kqNsJeoamSgc0YlckC+C9Egr7O8WcFdftZhHQbOuZ1NZfX2brFJEirM/kvm28wYzqxCMnPJiQ/ulekkRZZxK6y7O1XtxNeXK55rokU+fcmCB2Z5HbH64ysnqLn4+exDURIDzArkuY0ruTpwzGEOCoVCoVAoFAqFwh7+ATX923/I0qEaAAAAAElFTkSuQmCC\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the constraint         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral of f'^2 from -infinity to infinity = a\" style=\"width: 119.5px; height: 44px;\" width=\"119.5\" height=\"44\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves a jet in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x, y\" style=\"width: 24.5px; height: 18px;\" width=\"24.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plane emanating from a source at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(x,y) = (0,0)\" style=\"width: 87.5px; height: 19px;\" width=\"87.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. It can be solved by expressing the velocities \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the problem above, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(eta)\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is proportional to the streamfunction. The conditions at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"eta = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e define the line \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = 0\" style=\"width: 37px; height: 18px;\" width=\"37\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a line of symmetry; the transverse velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the shear stress are zero there. The condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e states that the streamwise velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 4/3\" style=\"width: 51.5px; height: 19px;\" width=\"51.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = lamjetODE(eta,a)\r\n  f = g(eta,a);\r\nend","test_suite":"%%\r\na = 4/3; \r\neta = 0:0.2:1;\r\nf_correct = [0 0.197375320224904 0.379948962255225 0.537049566998035 0.664036770267849 0.761594155955765];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 4/3; \r\neta = log(2);\r\nf_correct = 0.6;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\neta = pi;\r\nf_correct = 0.902552583791843;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.5;\r\neta = 1.5;\r\nf_correct = 0.572446107496431;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.75;\r\neta = -4:0;\r\nf_correct = [-0.823247562011528 -0.813902901498664 -0.766864130068021 -0.559711742572462 0];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = rand;\r\neta = 4*rand;\r\nassert(abs(lamjetODE(eta,a)+lamjetODE(-eta,a))\u003c1e-10)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2021-03-21T03:15:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-21T02:07:31.000Z","updated_at":"2021-03-22T00:42:37.000Z","published_at":"2021-03-21T02:24:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f’’’ + 2(f’2+ff”) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime\\\\prime + 2(f\\\\prime^2+ff\\\\prime\\\\prime) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f''(0) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f\\\\prime\\\\prime(0) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the constraint         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral of f'^2 from -infinity to infinity = a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty\\\\left[f\\\\prime(\\\\eta)\\\\right]^2d\\\\eta=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves a jet in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x, y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex, y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plane emanating from a source at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(x,y) = (0,0)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x,y) = (0,0)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It can be solved by expressing the velocities \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In the problem above, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(eta)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\eta)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is proportional to the streamfunction. The conditions at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e define the line \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a line of symmetry; the transverse velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the shear stress are zero there. The condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e states that the streamwise velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 4/3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 4/3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51808,"title":"Determine when snow started","description":"R.P. Agnew posed the following problem: It starts snowing in the morning and continues steadily throughout the day. A snowplow that removes snow at a constant rate starts plowing at noon. It plows 2 miles in the first hour and 1 mile in the second.  What time did it start snowing?\r\nWrite a function to solve this problem. The inputs will be the time the snowplow started (given as a string), two consecutive time intervals  and , and the distances  and  traveled in the first and second time intervals, respectively. Return the time that the snow started, rounded to the nearest minute, as a string.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 135.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 67.625px; transform-origin: 407px 67.625px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 367.333px 7.91667px; transform-origin: 367.333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eR.P. Agnew posed the following problem: It starts snowing in the morning and continues steadily throughout the day. A snowplow that removes snow at a constant rate starts plowing at noon. It plows 2 miles in the first hour and 1 mile in the second.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat time did it start snowing?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.625px; text-align: left; transform-origin: 384px 31.625px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.275px 7.91667px; transform-origin: 380.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem. The inputs will be the time the snowplow started (given as a string), two consecutive time intervals \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dt1\" style=\"width: 23px; height: 20px;\" width=\"23\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dt2\" style=\"width: 23px; height: 20px;\" width=\"23\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.675px 7.91667px; transform-origin: 60.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the distances \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dx1\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dx2\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 184.625px 7.91667px; transform-origin: 184.625px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e traveled in the first and second time intervals, respectively. Return the time that the snow started, rounded to the nearest minute, as a string.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tstr = snowplow(t0str,dx1,dx2,dt1,dt2)\r\n%  tstr = string denoting the time the snowplow started\r\n%  dt1  = first time interval\r\n%  dt2  = second time interval (immediately follows the first)\r\n%  dx1  = distance plowed during the first interval\r\n%  dx2  = distance plowed during the second interval\r\n\r\n  tstr = datestr((dt2-dt1)*dx2/dx1) + t0str;\r\n","test_suite":"%% Original problem\r\nt0str = '12:00';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 2; % miles\r\ndx2 = 1; % mile\r\ntstr_correct = '11:23';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '12:00';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 3.22; % km\r\ndx2 = 1.61; % km\r\ntstr_correct = '11:23';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '12:00';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 3.11; % km\r\ndx2 = 1.73; % km\r\ntstr_correct = '11:09';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '12:00';\r\ndt1 = 2; % hour\r\ndt2 = 1; % hour\r\ndx1 = 4.24; % km\r\ndx2 = 1.26; % km\r\ntstr_correct = '10:29';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '13:30';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 3.11; % km\r\ndx2 = 1.73; % km\r\ntstr_correct = '12:39';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '09:53';\r\ndt1 = 0.75; % hour\r\ndt2 = 1; % hour\r\ndx1 = 2.28; % km\r\ndx2 = 1.99; % km\r\ntstr_correct = '08:35';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-22T22:48:59.000Z","updated_at":"2026-01-02T12:00:02.000Z","published_at":"2021-05-22T22:52:55.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eR.P. Agnew posed the following problem: It starts snowing in the morning and continues steadily throughout the day. A snowplow that removes snow at a constant rate starts plowing at noon. It plows 2 miles in the first hour and 1 mile in the second.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eWhat time did it start snowing?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem. The inputs will be the time the snowplow started (given as a string), two consecutive time intervals \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dt1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta t_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dt2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta t_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the distances \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dx1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dx2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e traveled in the first and second time intervals, respectively. Return the time that the snow started, rounded to the nearest minute, as a string.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":53995,"title":"Solve an ODE: nonlinear third-order equation","description":"Write a function to solve the ordinary differential equation\r\n\r\non the domain  with , , and either  or . The input variable ord indicates the order (either 1 or 2) of the specified derivative. The function should return the values of  at the requested values of the independent variable .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 124px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 62px; transform-origin: 407.5px 62px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 176.35px 7.66667px; transform-origin: 176.35px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the ordinary differential equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'y'' = y'''\" style=\"width: 71px; height: 18px;\" width=\"71\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 32px; text-align: left; transform-origin: 384.5px 32px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.2917px 7.66667px; transform-origin: 46.2917px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eon the domain \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"0 \u003c= x \u003c= 1\" style=\"width: 62.5px; height: 18px;\" width=\"62.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.3333px 7.66667px; transform-origin: 16.3333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(0) = y0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(1) = y1\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.95px 7.66667px; transform-origin: 36.95px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and either \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'(0) = y'0\" style=\"width: 70px; height: 20px;\" width=\"70\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.66667px; transform-origin: 10.1083px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAoCAYAAADkIfgQAAAG1klEQVR4Xu1bOasmRRSd+QWDOpEYPFwCYUAZt0QDDRwXEMHAERMDcQ0V18BAHbfRyBUMXiDMCIIgKGqg4BK4gYJgoDOIgZEL4g/Qc6QP1Fev6tbt7qr+5vNVw6V576u+de+tU3er7r17+tUtsLAF9i48X5+uW2BPB10HweIW6KBb3OR9wg66joHFLbCJoHsZVvoEdLyStZ4En79Az1bi19kULNAadO9j/nNBl4F+T8hyHf73Buhh0GuO1RoDuMsHfr/ifrLA+wH8vgW6xyHD/3kI7fAM6ELQdxlFS2tatE9L0F2A2b8F/Qk6LwM6equbQQ86PA0BR16PGlqdjd/uBx0GPT2MuwP3n4Y5cobkUPL/2SFH0agbPOAPyH4a6ArQZwk9PGtaVL8l6Bi2HgEdyQDlDPz/twFIF+NueSOC6AlQzmNSURrkYxANdxNIAOM8X4DOAV0Pei9jFY77EXRDxuBFY274AEadd0Ffgy7J6FJaU5cJWoKO3uV0UA5QBNIxA5RSQGBg6MvlcSGw7sK4OFQz1H46MNyPeyrU82cPuF2G3cBB9PR3FzZmaU1darcCHcPciQKgqCQXueTlmGcwRDI3zF3KRRh+c/xoMHq7N4d5c7zoKRmad1thQfvQfjkv51nTtYLuTsz+VAFQXNxXQFaOJi9XKjSUi1ihQTuZhrG8HUMI80wL5C7jbtAg5WpW+uFZU5fKrTwdK5y3QbmKlOFuG2TlaFRAIdiqppSLcDxBnKtAabRXB6vcgnsuVCsUWwvgMu4GDWKkODjYOyd2aU3d6npAR29zPugHUJwL0eWeGf2m/MoCFL3JLwYopQCBcTWIuWHuUmjl76l8Ts+FeZ0FTo7/B5QrgNzGXeNAei5eqWqddvg7+u0r/H17Zjz5eNbUrW4OdBTsWhATS5bQuQVVWPsAA24FEZQUkIDLVYnkRf6e/hn5fwm6xtBIbZecjHo0BJ0Vhjmei0BdrHlzIqk/6F6ExMAYFB5e3HxXgQ4Ng5lTxxs/9PahJ2dEsZrtnjX1yPjfmJKnC6vClBLePMktUDBQLZWSV6Lbl6G9no4Js+U9xbNkn5Re9JJzr1KxY/EPgRXbQ20lOpK1eXKPUZVXUdFU05DAY8WTq3qmLoA8U6lxHILOyv0ohwBRAp02k8c+sX6UZ+71ERjMqZ5VqTMCxd5aVejaclaPUeVxaMjU7mh1djkFdLlOOmUP9SiBTnmixW8usFo+ryYu54grddnBs/ZNZPROrLyJuRB3TlhQcFeFJwC1BJ0CulrhddNBpxYI1yKu1Bm5bgTxvpbLC7owTwhDWEsFvKCbUr2WcibPwfdaFmzEpCryYl1bOQl60McD+RjGmRrtqKC9oMvtnFYKUHYv6MKc0+vpSkm0PLvXPiEW1lW9xnhMRadWTkJN/A8hxL1DJJSj2pFnjzGqdo6qSTK9srGbZuJfql6VGNPolgcLPWIpiWYxcCnIqnBzTmfd1avkCqMTj//4ziBffGiRCqnwinNgYoZdj5Uicwzo4p3DNzJK56YjokFyKD0pqdQvUwVLBXPHV5I/1fqJJ/fOmxL6VKheKVcYnbjJDoC2QLXfGQwLtLho0bqsgHEM6LRzWPnRjXLxrHPTuYDj897WRdj4TVWcoWGsEMw5NbbUqqmhX2seYXRio986c54qi2yfargruqykM2NAFy6s5VGmCp96Tueqpf4bn1WbIBVipXyqbxXP6znvraljS15hD9M6b54jg2xrgW7F7mNAR8GUr5RyojlKxM963kbRM/KM3FkvgNjakYcOj+os+Vo1u2vaxMvLAoSXR2mcB3QrgJwCulJiXxJy7O9U6iGQN6nXufFFw0Tf4P45yDoLlkwKra28wljd544XIFhIlL4TmTpXU9DpdaSl3zPT+e9zsIrn452pxuNzOjQvFS5z5ljyWYbXbVCtL+dSsiuSWOF1kqdjW4IPtq5WcwtCwL8Dyn3gU2MhdRjONpD1AU+NuZbgwRyXB/u1q9VY9iaFBD0Nd8xjzhDVyqBM8G8DtfBC0vH5xl6hlW1ivrTVfaDaL2Hk5FeuH6drKmRWaoBUTqfF5ZsOb4FYDb4Oah3aPAvC8LfVYPeyeKj5AbdHl1pjuGFeGpi9iPtZoNKXc7XmFh+rOcxCcOW9vhTowjKbTE+1pJqbYl/FTUAgfw/yFBq1F6sGv7CVRX6plzJqzGPx0DFY+MKtcr0dnY4U6Jjb8EMYIvQoqFXV09oQu4m/CqBtKN2yaLBsqg/deWfHgBUzPe+Oj7bHtkx200J2XRtZoIOukWE727wFOug6Oha3QAfd4ibvE3bQdQwsboEOusVN3ifsoOsYWNwC/wJfybU4YZYqZwAAAABJRU5ErkJggg==\" alt=\"y''(0) = y''0\" style=\"width: 78.5px; height: 20px;\" width=\"78.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 7.66667px; transform-origin: 61.4583px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The input variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.66667px; transform-origin: 11.55px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; \"\u003eord\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.5083px 7.66667px; transform-origin: 31.5083px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e indicates the order (either 1 or 2) of the specified derivative. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x)\" style=\"width: 30px; height: 18.5px;\" width=\"30\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 7.66667px; transform-origin: 84.4px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the requested values of the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solve3ordNLODE(x,y0,y1,yd,ord)\r\n%  x = requested values of the independent variable\r\n%  y = values of function at the requested values of x\r\n%  y0, y1 = values of the function at x = 0 and x = 1\r\n%  yd = value of either y'(0) or y''(0)\r\n%  ord = 1 if yd is the first derivative or 2 if it is the second derivative\r\n\r\n   y = A1*exp(r1*x)+A2*exp(r2*x)+A3*exp(r3*x);\r\nend","test_suite":"%%\r\ny0 = 2.231252941;\r\ny1 = 6.297567308;\r\nyd = 1.557407725;\r\nord = 1;\r\nx  = [0.3 0.5 0.8];\r\ny_correct = [2.790588219 3.308319582 4.544300275];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 2.231252941;\r\ny1 = 6.297567308;\r\nyd = 1.71275941;\r\nord = 2;\r\nx  = [0.3 0.5 0.8];\r\ny_correct = [2.790588219 3.308319582 4.544300275];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 3.287682072;\r\ny1 = 9.107569130;\r\nyd = 1.154700538;\r\nord = 1;\r\nx  = 0.25:0.25:0.75;\r\ny_correct = [3.669824691 4.306714708 5.456245713];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 3.287682072;\r\ny1 = 9.107569130;\r\nyd = 8/3;\r\nord = 2;\r\nx  = 0.25:0.25:0.75;\r\ny_correct = [3.669824691 4.306714708 5.456245713];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 3.693147181;\r\ny1 = 3.046411846;\r\nyd = -2;\r\nord = 1;\r\nx  = 0.2:0.2:0.8;\r\ny_correct = [3.36426198 3.152361229 3.034571214 3.000213221];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 0.722781494;\r\ny1 = 2.637280183;\r\nyd = 1.030077779;\r\nord = 2;\r\nx  = 0.2:0.2:0.8;\r\ny_correct = [0.950884887 1.231252941 1.581096155 2.030246566];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-02-07T03:08:13.000Z","updated_at":"2022-02-07T13:11:31.000Z","published_at":"2022-02-07T03:11:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the ordinary differential equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'y'' = y'''\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime y\\\\prime\\\\prime = y\\\\prime\\\\prime\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eon the domain \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"0 \u0026lt;= x \u0026lt;= 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \\\\le x \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and either \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(0) = y\\\\prime\\n_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y''(0) = y''0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime\\\\prime(0) = y\\\\prime\\n\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The input variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eord\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e indicates the order (either 1 or 2) of the specified derivative. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the requested values of the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51783,"title":"Solve an ODE: equation B","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 213.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 106.625px; transform-origin: 407px 106.625px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.358px 7.91667px; transform-origin: 211.358px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation:  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\" style=\"width: 181.5px; height: 39px;\" width=\"181.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.0083px 7.91667px; transform-origin: 35.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and either \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'(x0) = y'0\" style=\"width: 74px; height: 20px;\" width=\"74\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1167px 7.91667px; transform-origin: 38.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Along with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.275px 7.91667px; transform-origin: 25.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, one of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eyp0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.708px 7.91667px; transform-origin: 109.708px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will be assigned numerical values, and the other will be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 7.91667px; transform-origin: 80.9px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.483px 7.91667px; transform-origin: 194.483px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne of the applications for this equation is in groundwater. For \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 1\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p = 0\" style=\"width: 38px; height: 18px;\" width=\"38\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.567px 7.91667px; transform-origin: 136.567px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.975px 7.91667px; transform-origin: 141.975px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the normalized distance from the well, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.94167px 7.91667px; transform-origin: 8.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEB(x,coeff,Xbc,bc)\r\n%  y = values of the solution\r\n%  x = values of the independent variable where the solution is requested\r\n%  coeff = [a p], the two parameters in the ODE\r\n%  Xbc = values of x where the boundary conditions are specifified\r\n%  bc = [y0 yp0 y1] = boundary values (see description)\r\n\r\n  y = f(x,coeff,Xbc,bc);\r\nend","test_suite":"%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [1 NaN 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [1 0.505461057363874 0.265959532078956 0.140787844664873 0.071276733472728 0.029333752731051 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [0 NaN 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000035290165611 0.000065476121971 0.000315436935125 0.000552779648598 0.000757412623201 0.003086031722519 0.012031119481478 0.040129255417251 0.089700339919646];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [4 NaN -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [2.974028994378906 1.041176175714286 0.308462205553512 -0.076175488441917 -0.426089583908447 -0.952692417292867];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [2 4];\r\nXbc = [1 Inf];\r\nbc = [3 NaN 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [3 0.005688558853080 0.000051725255081 0.000000653418086 0.000000009396692];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n%-------------------------------\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [NaN 1 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-0.696760997897786 -0.352185550727323 -0.185310228971762 -0.098095479140575 -0.049662847941353 -0.020438614824974 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n \r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [NaN 0 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000053997354167 0.000075728700374 0.000316920386259 0.000553525214653 0.000757922298088 0.003086129240342 0.012031140116004 0.040129260776539 0.089700342119009];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [NaN 4 -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.047695617799804 -0.816404083826666 -0.431398160565887 -0.374418157507686 -0.532801281305956 -0.958228725044217];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 Inf];\r\nbc = [NaN 3 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.098451806781317 -0.173147136186896 -0.018397012773047 -0.002117248575316 -0.000253600440751];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-16T01:10:22.000Z","updated_at":"2025-05-09T06:39:10.000Z","published_at":"2021-05-16T01:19:31.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation:  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} +\\\\frac{1}{x} \\\\frac{dy}{dx}-\\\\left(a^2+\\\\frac{p^2}{x^2}\\\\right)y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and either \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(x0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(x_0) = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Along with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, one of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eyp0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will be assigned numerical values, and the other will be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne of the applications for this equation is in groundwater. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the normalized distance from the well, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":45976,"title":"Evaluate the Struve function","description":"The \u003chttps://en.wikipedia.org/wiki/Struve_function Struve function\u003e *H*_a(x) is a solution to an inhomogeneous form of Bessel's equation:\r\n\r\n\u003c\u003chttps://wikimedia.org/api/rest_v1/media/math/render/svg/246a7bab900d24f188a7edcec59042852728d747\u003e\u003e\r\n\r\nThe Struve function appears in applications of optics and loudspeaker design. I encountered it in developing a model of turbulence in a strongly stratified fluid (e.g., a lake with a large temperature difference). \r\n\r\nEvaluate the Struve function *H*_p(x) for given values of the order p and argument x. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 151.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 75.7917px; transform-origin: 407px 75.7917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9083px; text-align: left; transform-origin: 384px 10.9083px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0583px 7.79167px; transform-origin: 12.0583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Struve_function\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eStruve function\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"H_p(x)\" style=\"width: 40px; height: 20px;\" width=\"40\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 190.383px 7.79167px; transform-origin: 190.383px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a solution to an inhomogeneous form of Bessel's equation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38.95px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.475px; text-align: left; transform-origin: 384px 19.475px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5333px 7.79167px; transform-origin: 15.5333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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alt=\"Differential equation from the Wikipedia link\" style=\"width: 271.5px; height: 39px;\" width=\"271.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 370.333px 7.79167px; transform-origin: 370.333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe Struve function appears in applications of optics and loudspeaker design. I encountered it in developing a model of turbulence in a strongly stratified fluid (e.g., a lake with a large temperature difference).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9083px; text-align: left; transform-origin: 384px 10.9083px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.5167px 7.79167px; transform-origin: 87.5167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEvaluate the Struve function\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"H_p(x)\" style=\"width: 40px; height: 20px;\" width=\"40\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90.625px 7.79167px; transform-origin: 90.625px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for given values of the order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ep\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.0667px 7.79167px; transform-origin: 47.0667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and argument \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function Hp = StruveFn(p,x)\r\n  Hp = f(x,p);\r\nend","test_suite":"%%\r\nx = 0;\r\np = randi(8,1);\r\nHp_correct = 0;\r\nassert(isequal(StruveFn(p,x),Hp_correct))\r\n\r\n%%\r\nx = 0.4;\r\np = 0;\r\nHp_correct = 0.2501497138634162;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = 10;\r\np = 0;\r\nHp_correct = 0.1187436836875042;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = rand(1);\r\np = 1/2;\r\nHp_correct = sqrt(2/(pi*x))*(1-cos(x));\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = 4.2;\r\np = 1;\r\nHp_correct = 1.036818631956923;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%% \r\nx = pi^2;\r\np = 3;\r\nHp_correct = 4.10841348624688;\r\nassert(abs(StruveFn(p,x)-Hp_correct)/Hp_correct\u003c1e-6)\r\n\r\n%%\r\nx = rand(1);\r\np = 1;\r\nHp_approx = 2/pi - besselj(0,x) + (16/pi-5)*sin(x)/x + (12-36/pi)*(1-cos(x))/x^2;\r\nassert(abs(StruveFn(p,x)-Hp_approx)/Hp_approx\u003c0.002)","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":"2021-01-02T18:43:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-06-21T16:48:35.000Z","updated_at":"2026-01-09T13:52:55.000Z","published_at":"2020-06-21T17:11:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Struve_function\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eStruve function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"H_p(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\bf H}_p(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a solution to an inhomogeneous form of Bessel's equation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"Differential equation from the Wikipedia link\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^2\\\\frac{d^2y}{dx^2}+x\\\\frac{dy}{dx}+(x^2-p^2)y=\\\\frac{4(x/2)^{p+1}}{\\\\sqrt{\\\\pi}\\\\Gamma(p+1/2)}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Struve function appears in applications of optics and loudspeaker design. I encountered it in developing a model of turbulence in a strongly stratified fluid (e.g., a lake with a large temperature difference).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEvaluate the Struve function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"H_p(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{\\\\bf H}_p(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for given values of the order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and argument \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51625,"title":"Solve an ODE: diffusion problem 2","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a(f+eta f') = 0\" style=\"width: 130px; height: 19px;\" width=\"130\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral(f,{eta,-inf,inf}) = 1\" style=\"width: 78px; height: 44px;\" width=\"78\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion2ODE(eta,a)\r\n%  eta = independent variable\r\n%  a   = positive constant\r\n  f = a*sqrt(eta);\r\nend","test_suite":"%%\r\na = 1/2; \r\neta = 0:0.2:1;\r\nf_correct = [0.282094791773878 0.279287901697234 0.271033696776216 0.257815227404741 0.240385324709827 0.219695644733861];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2; \r\neta = 0.5;\r\nf_correct = 0.439391289467722;\r\nassert(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13)\r\n\r\n%%\r\na = 1;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.396080211793656 0.388528585315836 0.359548380672770 0.292234407541508 0.239309153606614];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2;\r\neta = 1:4;\r\nf_correct = [0.207553748710297 0.010333492677046 0.000069626525973 0.000000063491173];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 3/4;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [0.345377564870647 0.334028296534584 0.011822159682599];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 1;\r\neta = rand/120;\r\nf_correct = polyval([1/10321920 0 -1/645120 1/46080 0 -1/3840 0 1/384 0 -1/48 0 1/8 0 -1/2 0 1]./(sqrt(2*pi)),eta);\r\nf = diffusion2ODE(eta,a);\r\nassert(all(abs(f-f_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-01T14:31:00.000Z","updated_at":"2021-05-01T14:34:52.000Z","published_at":"2021-05-01T14:34:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a(f+eta f') = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime+a(f+\\\\eta f\\\\prime)=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral(f,{eta,-inf,inf}) = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty fd\\\\eta=1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51745,"title":"Solve an ODE: equation A","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith  and . The function should return the values of  at the specified values of . One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.6px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.3px; transform-origin: 407px 59.3px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 8px; transform-origin: 209.417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.3px; text-align: left; transform-origin: 384px 18.3px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y\u0026quot;-xy = 0\" style=\"width: 74px; height: 36.5px;\" width=\"74\" height=\"36.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 8px; transform-origin: 14.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 8px; transform-origin: 128.742px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 8px; transform-origin: 80.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.2333px 8px; transform-origin: 62.2333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEA(x,x0,y0,x1,y1)\r\n  y(x0) = y0; \r\n  y(x1) = y1; \r\n  y = f(x,x0,y0,x1,y1);\r\nend","test_suite":"%%\r\nx0 = -3; \r\ny0 = 1;\r\nx1 = 3;\r\ny1 = 0.2;\r\nx  = linspace(x0,x1,7);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [1 -0.608544735138462 -1.416513846161609 -0.930561687980622 -0.339539877942166 -0.041385541117497 0.2];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -5; \r\ny0 = 2;\r\nx1 = 5;\r\ny1 = 0;\r\nx  = linspace(x0,x1,11);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [2 -0.400646543833150 -2.159956361501116 1.296651996575315 3.053707895612597 2.024329503005815 0.771421025528832 0.199130370987114 0.037568752980204 0.005346964905914 0];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -4; \r\ny0 = -1;\r\nx1 = 2;\r\ny1 = 0.3;\r\nx  = linspace(x0,x1,6);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [-1 -4.088820829832713 5.996215644909088 6.297137060461841 2.304927532867409 0.3];\r\nassert(all(abs(y-y_correct) \u003c 1e-13))\r\n\r\n%%\r\nx0 = -7; \r\ny0 = 0;\r\nx1 = 3;\r\ny1 = 0.1;\r\nx  = linspace(x0,x1,6);\r\ny  = solveODEA(x,x0,y0,x1,y1);\r\ny_correct = [0 -0.004972738967217 0.002891447704152 -0.005345046731767 0.007070410943182 0.1];\r\nassert(all(abs(y-y_correct) \u003c 1e-14))\r\n\r\n%% anti-cheating--product of two values\r\nx0 = -2; \r\ny0 = 1;\r\nx1 = 2;\r\ny1 = 0.1;\r\nx  = [-1 1];\r\nz  = prod(solveODEA(x,x0,y0,x1,y1));  \r\nz_correct = 1.336786968358133;\r\nassert(all(abs(z-z_correct) \u003c 1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-15T13:16:15.000Z","deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":"2021-05-14T12:28:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-11T04:16:02.000Z","updated_at":"2025-09-02T13:25:13.000Z","published_at":"2021-05-11T04:19:47.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot;-xy = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} – x y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. One application of this equation involves the propagation of internal gravity waves in a fluid with variable density gradient.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54670,"title":"Solve an ODE: precocious pair’s porcine pursuit","description":"In our previous encounters with Matilda and Labrun, the scintillating siblings collected candy wrappers, amused others with card tricks, and found interesting relations involving house numbers on their street. \r\nBut now their pet pig has run away, and the pair must catch her! They start a distance  away from the pig, which runs at speed  in a direction perpendicular to the line connecting the initial positions of the pig and the siblings. Matilda and Labrun run at speed , always in a direction pointing at the current position of the pig. \r\nWrite a function that takes the distance  and the two speeds and returns the time required for Matilda and Labrun to catch their pet. Return Inf if the pair will not catch the pig, and please ignore the impracticality of reporting times to the nearest microsecond. \r\nThis problem is adapted from a problem in Advanced Mathematical Methods for Scientists and Engineers by Bender and Orzsag.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 237.45px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 118.725px; transform-origin: 407px 118.725px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.133px 7.79167px; transform-origin: 236.133px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn our previous encounters with Matilda and Labrun, the scintillating siblings \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/53004\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ecollected candy wrappers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/51451\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eamused others with card tricks\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.79167px; transform-origin: 17.5px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/51251\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003efound interesting relations involving house numbers on their street\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.292px 7.79167px; transform-origin: 266.292px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBut now their pet pig has run away, and the pair must catch her! They start a distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ed\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104.233px 7.79167px; transform-origin: 104.233px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e away from the pig, which runs at speed \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eV\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 336.5px 7.79167px; transform-origin: 336.5px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in a direction perpendicular to the line connecting the initial positions of the pig and the siblings. Matilda and Labrun run at speed \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 197.208px 7.79167px; transform-origin: 197.208px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, always in a direction pointing at the current position of the pig. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.45px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.725px; text-align: left; transform-origin: 384px 31.725px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.383px 7.79167px; transform-origin: 122.383px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the distance \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ed\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 254.775px 7.79167px; transform-origin: 254.775px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the two speeds and returns the time required for Matilda and Labrun to catch their pet. Return \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.79167px; transform-origin: 11.55px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 11.55px 8.25px; transform-origin: 11.55px 8.25px; \"\u003eInf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 311.95px 7.79167px; transform-origin: 311.95px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e if the pair will not catch the pig, and please ignore the impracticality of reporting times to the nearest microsecond. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 133.033px 7.79167px; transform-origin: 133.033px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis problem is adapted from a problem in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.492px 7.79167px; transform-origin: 194.492px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eAdvanced Mathematical Methods for Scientists and Engineers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.79167px; transform-origin: 47.4583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eby Bender and Orzsag.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function t = pigPursuit(d,V,v)\r\n%  d = initial distance between pig and M\u0026L. The pig runs perpendicular to the line connecting the initial positions\r\n%  V = pig speed\r\n%  v = M\u0026L speed\r\n%  t = time till capture\r\n\r\n  y = hypot(d/V,d/v);\r\nend","test_suite":"%%\r\nd = 5;                      %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 4.1;                    %  M\u0026L speed (m/s)\r\nt_correct = 25.308642;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 5;                      %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 4.5;                    %  M\u0026L speed (m/s)\r\nt_correct = 5.294118;       %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 5;                      %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.1;                    %  M\u0026L speed (m/s)\r\nt_correct = 25.247525;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 5;                      %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.2;                    %  M\u0026L speed (m/s)\r\nt_correct = 12.745098;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 5;                      %  M\u0026L speed (m/s)\r\nt_correct = 5.555556;       %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 4.3;                    %  M\u0026L speed (m/s)\r\nt_correct = 17.269076;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.5;                    %  M\u0026L speed (m/s)\r\nt_correct = 10.476190;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 10;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 6;                      %  M\u0026L speed (m/s)\r\nt_correct = 5.454545;       %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 5;                      %  M\u0026L speed (m/s)\r\nt_correct = 11.111111;      %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 4;                      %  Pig speed (m/s)\r\nv = 6;                      %  M\u0026L speed (m/s)\r\nt_correct = 6;              %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.01;                   %  M\u0026L speed (m/s)\r\nt_correct = 1000.999001;    %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 20;                     %  Distance (m)\r\nV = 5;                      %  Pig speed (m/s)\r\nv = 5.001;                  %  M\u0026L speed (m/s)\r\nt_correct = 10000.9999;     %  Time (s)\r\nassert(abs(pigPursuit(d,V,v)-t_correct)\u003c1e-6)\r\n\r\n%%\r\nd = 100*rand;\r\nV = 6*rand;\r\nv = V;\r\nassert(isinf(pigPursuit(d,V,v)))\r\n\r\n%%\r\nd = 100*rand;\r\nV = 6*rand;\r\nv = V*rand;\r\nassert(isinf(pigPursuit(d,V,v)))\r\n\r\n%%\r\nfiletext = fileread('pigPursuit.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'import'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2022-05-24T14:38:30.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2022-05-24T14:38:30.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-05-24T14:03:33.000Z","updated_at":"2022-05-24T14:38:30.000Z","published_at":"2022-05-24T14:05:54.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn our previous encounters with Matilda and Labrun, the scintillating siblings \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/53004\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ecollected candy wrappers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/51451\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eamused others with card tricks\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/51251\\\"\u003e\u003cw:r\u003e\u003cw:t\u003efound interesting relations involving house numbers on their street\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBut now their pet pig has run away, and the pair must catch her! They start a distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"d\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ed\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e away from the pig, which runs at speed \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"V\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eV\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in a direction perpendicular to the line connecting the initial positions of the pig and the siblings. Matilda and Labrun run at speed \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, always in a direction pointing at the current position of the pig. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the distance \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"d\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ed\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the two speeds and returns the time required for Matilda and Labrun to catch their pet. Return \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if the pair will not catch the pig, and please ignore the impracticality of reporting times to the nearest microsecond. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem is adapted from a problem in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eAdvanced Mathematical Methods for Scientists and Engineers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eby Bender and Orzsag.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51815,"title":"Solve an ODE: equation C","description":"Write a function to solve the following ordinary differential equation: \r\n\r\nwith  and . The parameter  is a constant. The function should return the values of  at the specified values of .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 59.2917px; transform-origin: 407px 59.2917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 209.417px 7.91667px; transform-origin: 209.417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\" style=\"width: 179.5px; height: 37.5px;\" width=\"179.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.725px 7.91667px; transform-origin: 51.725px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169.967px 7.91667px; transform-origin: 169.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.95px 7.91667px; transform-origin: 50.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEC(x,a,Xbc,bc)\r\n%  y   = values of the function\r\n%  x   = independent variable\r\n%  a   = parameter in the equation\r\n%  Xbc = [x0 x1], values of x where the boundary conditions are specified\r\n%  bc  = [y0 y1], values of the function at x0 and x1, respectively\r\n\r\n   y = f(x,a,Xbc,bc);\r\nend","test_suite":"%% \r\nx   = [-1/3 -1/4 0 1/4 1/3];\r\na   = 1;\r\nXbc = [-1/2 1/2];\r\nbc  = [2 4];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [2.599319657044238 2.854101966249684 3.464101615137754 3.854101966249684 3.932652990377571];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx   = [-0.6 -0.2 -0.05 0.12 0.2];\r\na   = sqrt(3);\r\nXbc = [-0.7 0.3];\r\nbc  = [0 1];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [0.237055225759061 0.877546669526703 0.995431932094838 1.046546024897035 1.039090864471318];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx   = [-0.8 -0.6 -0.3 -0.15 0.1 0.25 0.375];\r\na   = 2;\r\nXbc = [-0.8 0.4];\r\nbc  = [-1 3];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [-1 1.68647951290722 4.138417234449975 4.687455882945745 4.630189304757032 4.024530246664777 3.199461161409147];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%% \r\nx   = [-0.7 -0.45 -0.2 0.05 0.3 0.55 0.8];\r\na   = pi;\r\nXbc = [-0.9 0.9];\r\nbc  = [-1 1];\r\ny   = solveODEC(x,a,Xbc,bc);\r\ny_correct = [1.764854605944604 2.706629373469937 1.6090059417224112 -0.4259046519843118 -2.225041323571773 -2.630850806938173 -0.6162122684969365];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\na   = 5;\r\nXbc = [-1/2 1/2]; \r\nbc  = [-1 1];\r\ny   = 1;\r\nx   = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na   = 5;\r\nXbc = [-1/2 1/2]; \r\nbc  = [-1 1];\r\ny   = 1;\r\nx   = fzero(@(z) solveODEC(z,a,Xbc,bc)-y,0);\r\nx_correct = 0.104528463267653;\r\nassert(abs(x-x_correct)\u003c1e-13)\r\n\r\n%%\r\na   = 2*randi(4)+1;\r\nXbc = [-3/4 3/4]; \r\nbc  = [-2 2];\r\nx   = rand-3/4;\r\nym  = solveODEC(-x,a,Xbc,bc);\r\nyp  = solveODEC(x,a,Xbc,bc);\r\nassert(abs(ym+yp)\u003c1e-13)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-25T04:18:45.000Z","updated_at":"2021-05-25T04:23:56.000Z","published_at":"2021-05-25T04:23:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(1-x^2) y\u0026quot; - x y' + a^2 y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(1-x^2)\\\\frac{d^2y}{dx^2} -x \\\\frac{dy}{dx}+a^2 y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":231,"title":"Differential equations I","description":"Given a function handle |f| an initial condition |y0| and a final time |tf|, solve numerically the differential equation\r\n\r\n  dy/dt = f(y)\r\n\r\nfor the function |y(t)| between |t=0| and |t=tf|. Give as a result |res=y(tf)|.\r\n\r\nExample:\r\n\r\n   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\r\n\r\n =\u003e y(tf) = 1/e = 0.367879441171442\r\n\r\nRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].","description_html":"\u003cp\u003eGiven a function handle \u003ctt\u003ef\u003c/tt\u003e an initial condition \u003ctt\u003ey0\u003c/tt\u003e and a final time \u003ctt\u003etf\u003c/tt\u003e, solve numerically the differential equation\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003edy/dt = f(y)\r\n\u003c/pre\u003e\u003cp\u003efor the function \u003ctt\u003ey(t)\u003c/tt\u003e between \u003ctt\u003et=0\u003c/tt\u003e and \u003ctt\u003et=tf\u003c/tt\u003e. Give as a result \u003ctt\u003eres=y(tf)\u003c/tt\u003e.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e   f = @(x) -x;\r\n   tf= 1;\r\n   y0= 1;\u003c/pre\u003e\u003cpre\u003e =\u003e y(tf) = 1/e = 0.367879441171442\u003c/pre\u003e\u003cp\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/p\u003e","function_template":"function res = deqnsolve(f,y0,tf)\r\n  res = 0;\r\nend","test_suite":"%% \r\nf = @(x) -x;\r\ntf =1;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(-1)) \u003c 1e-5)\r\n\r\n%% \r\nf = @sin;\r\ntf =1;\r\ny0 =1/2;\r\nassert(abs(deqnsolve(f,y0,tf)-2*acot(exp(-1)*cot(1/4))) \u003c 1e-5)\r\n\r\n%% \r\nf = @(x) 1/(x+1);\r\ntf =6;\r\ny0 =1;\r\nassert(abs(deqnsolve(f,y0,tf)-3) \u003c 1e-5)\r\n\r\n%% a randomized one\r\na = rand*0.9;\r\nf = @(x) x-a*x^2;\r\ntf = rand+1.5;\r\ny0=1;\r\nassert(abs(deqnsolve(f,y0,tf)-exp(tf)/(1-a+a*exp(tf))) \u003c 1e-5)","published":true,"deleted":false,"likes_count":3,"comments_count":5,"created_by":203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":190,"test_suite_updated_at":"2012-02-02T15:10:17.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-02T14:57:14.000Z","updated_at":"2026-03-13T19:43:52.000Z","published_at":"2012-02-02T15:20:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a function handle\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e an initial condition\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and a final time\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, solve numerically the differential equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[dy/dt = f(y)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor the function\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey(t)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et=tf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Give as a result\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eres=y(tf)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   f = @(x) -x;\\n   tf= 1;\\n   y0= 1;\\n\\n =\u003e y(tf) = 1/e = 0.367879441171442]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRemarks: aim at a relative precision of around 1e-6. The function is analytic in the interval [0,1].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":50609,"title":"Solve an ODE: equidimensional equation","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 140.45px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 70.225px; transform-origin: 407px 70.225px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 152.375px 7.79167px; transform-origin: 152.375px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSolve the following ordinary differential equation: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.1833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5917px; text-align: left; transform-origin: 384px 18.5917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x^2 y''(x) + a x y'(x) + b y(x) = 0\" style=\"width: 144px; height: 37px;\" width=\"144\" height=\"37\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64.2667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 32.1333px; text-align: left; transform-origin: 384px 32.1333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.79167px; transform-origin: 14.3917px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x_0) = y_0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJUAAAAoCAYAAAD39rjkAAAEZElEQVR4nO2ba5GzMBiFjwccYAADKKgCHNQBDmoBDZWwHrCAhlro9yOc4SWbKwm3b/PM7MxO6UI4vPewQKFQKBQKhYtTA6jOXkTh/+IHQH/2IgrHUR9w/m+m61QoEc9Gjf2fpZUKQAugg4oePztf7wlgTPj7FsADy1pPE+7CNFCOm6JzEj2A97yIL4DXzteboAxrCx3U+rjW00S7ONRoOHMRLZYH9djxOvSg1OhylAPclQlKn+bMRXTzIj7Yt0Z5QUXFFFiTfaGcobCGjpuqczLDvJC966kJyoBTOMoB7soTF4hSgHpAX+zb5j+QxxBY/+3tAHdlxAWiFMPl3ulkQJ7C8QgHuCsVlDaHR/AGKkT28+89lnQiFydbd1NhzWMhYbaazx/bBNRYRh0PrBsK03Xlvem08+d3qMO26s/u+DBaqJTBlv4x/85O4S2+xxmQq8visQl+z+jm74VSQ0W1z3ztFiqsj/jtAMAyFuHxL9Zji0F8PgasF1h0SP2J7XRT9T9sbsfiTReUUUp/CIQ3ZqqFYgrmN8I9qJ3P+cHaC2WUstUMjIiy5uIDesx/F5o2pTYpPykFcy79s8NBmCmiSOFsIZbHTV1byASeI4AQcXm9D36nKWlUrg6ShfyEJRpvET1XpEp54Dn0zw6t2VaEMy24UhM931RkD/BPx0O3ZRq4i3B5L64Q/xTfm3CB1jqRVP2zIlOBLfVMnuPA2vCk11XzZ74cPiLsxhnqbQY4eI4T2c2euk2RiVT9TdRYGpqoxkUWp6YLy8m0qyuzRbse/jopdFtGXsOW2kIcgPic6U6k6G+ix1Je1IiYa5kKVh3WWr5CTxofI06ol4Ruy7gKUmBdW/g8S6bRmI5T56zuTydFfx3qKDMHz+81UFnUmuqTEKOT6GOHV8giEL4tw7Xa1uIzOlJhPXpI2by+QvdHtupvO4+uyQilrVMrKYgptclXXWh0rhPKeoaRwNfV0Ct835MOYBJKFt40OtvLeQPUvcm6aute4xW6P7JFfx1qos/45Pmdta/LqDiIlOmkgnpgNq+SeX1E2IMK3ZZxGVWDddShAwyGNegDVr1r4rnuyBb9becwaUB7cT4v6akceFZQD23C+kU3iu1aqDxf6FwkZluGBiAHns95rdJBOizG2sxrGbAUnzLaynkVDe6u44Ut+utQR5dReZ1Opji5oArrlPJBeN2jT7ltdDCHWRtyPfqMqdE+ZzSTxmYalnae43cjRn8TWYwKWDYd9Y1HGlaH8NzszbmCmG0Zws3tHuaU/cQ6EvEeXB1W5zl+J2L0N5HNqHLxRPg7Onz94q6p5orE6G+DjZPLqA57D6tF3P4Za6FCHmL1t8F5lOnZBHV/uWgQX+CGbssU/GzR34VrTrVbdmFX+IOlY4opcHP+o+hfJFV/H66J+m77pLJj3OIhL5R3x1NI1T+EF5ZumDsQnArsguwKt5Djv2X+Mqn6h1LDPBW4JG+Uf5sqFAqFQqFQKBQKhULhRvwDMzwfDT/kVYEAAAAASUVORK5CYII=\" alt=\"dy/dx = y'_0\" style=\"width: 74.5px; height: 20px;\" width=\"74.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 7.79167px; transform-origin: 9.71667px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x = x0\" style=\"width: 40.5px; height: 20px;\" width=\"40.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 55.225px 7.79167px; transform-origin: 55.225px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The parameters \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 160.242px 7.79167px; transform-origin: 160.242px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are constants, and the value of the function and its derivative at the point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.79167px; transform-origin: 7.7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 7.7px 8.25px; transform-origin: 7.7px 8.25px; \"\u003ex0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 52.9px 7.79167px; transform-origin: 52.9px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e are specified as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.79167px; transform-origin: 7.7px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 7.7px 8.25px; transform-origin: 7.7px 8.25px; \"\u003ey0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.79167px; transform-origin: 11.55px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 11.55px 8.25px; transform-origin: 11.55px 8.25px; \"\u003eyp0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 205.767px 7.79167px; transform-origin: 205.767px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, respectively. Your function should return the value of the function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.00833px 7.79167px; transform-origin: 9.00833px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = equidimODE(x,a,b,y0,yp0,x0)\r\n%  a,b = parameters in the ODE\r\n%  x   = point at which the solution y is to be evaluated\r\n%  x0  = point at which the conditions are specified\r\n%  y0  = value of the solution at x = x0\r\n%  yp0 = value of the derivative at x = x0\r\n\r\n y = f(x,a,b,y0,yp0);\r\nend","test_suite":"%%\r\na  = 2; b  = -1; x   = 4;\r\nx0 = 1; y0 = 1;  yp0 = 0;\r\ny_correct = 1.733830915729880;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = 3; b  = -1; x   = 4;\r\nx0 = 2; y0 = 1;  yp0 = 3;\r\ny_correct = 3.593733292875542;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = 1; b  = 1; x   = 4;\r\nx0 = 1; y0 = 1; yp0 = 0;\r\ny_correct = 0.183456974743302;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = 0.5; b  = 1; x   = 6;\r\nx0 = 0.2; y0 = 1; yp0 = -1;\r\ny_correct = -2.149237864206678;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = -3; b  = 4; x   = 5;\r\nx0 = 1;  y0 = 0; yp0 = 1;\r\ny_correct = 40.235947810852508;\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n\r\n%%\r\na  = rand; b  = 0; x   = 5;\r\nx0 = 1;    y0 = 1; yp0 = 1;\r\ny_correct = y0+(x0*yp0/(1-a))*((x/x0)^(1-a)-1);\r\ny = equidimODE(x,a,b,y0,yp0,x0);\r\nassert(abs(y-y_correct)\u003c1e-10)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-02-27T23:20:45.000Z","updated_at":"2024-12-09T20:16:25.000Z","published_at":"2021-02-28T00:00:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSolve the following ordinary differential equation: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x^2 y''(x) + a x y'(x) + b y(x) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^2 {d^2y\\\\over dx^2} + a x {dy\\\\over dx} + b y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x_0) = y_0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dy/dx = y'_0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003edy/dx = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x = x0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = x_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The parameters \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are constants, and the value of the function and its derivative at the point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e are specified as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eyp0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e, respectively. Your function should return the value of the function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":50928,"title":"Solve an ODE: separable equation","description":"Solve the following ordinary differential equation:\r\n\r\nwith the initial condition .The test suite will ask for the value of the solution  at point . Functions such as ode45, ode23, and ode15S are not allowed. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 118px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 59px; transform-origin: 407.5px 59px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 150.433px 7.66667px; transform-origin: 150.433px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSolve the following ordinary differential equation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 18px; text-align: left; transform-origin: 384.5px 18px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAABICAYAAAA6cC0vAAALoUlEQVR4Xu2dWeh9VRXH9VXILH0SkT9aoKAojmD5YA+VCaKYE/bwf5AcEQwbnIigQS0VxNRCwQdHMgJRqQQFC0HNUDAUHBAMn5zAhx5rfX6er662ezzDvffc/zmwuP//755z9tprf/ea9tr77r3Xci0SMAnsvUhhkQASWICw4GBHAgsQFiAsQFgw8JkEFo2wGWi4w9h4xuihkdj5s73nNqMnat+3AKFWUuX7vmO3XGG0f3frH+zzHqP3C4+ODQI1BxjurQXXmEC4yBr9XVleW3nHj6xXNxpd3MngDfs81P0/1WlA8KHRtRNIBUAChiuN/l56/xAg/MJefqzRV7pOP2yf55Ua3MLv0QSPG/n+f2D//1IBCMjq50YnGpW0Rl+xfd0efNToq6U2hgAB5iQE/q3Z0JfpuT6n2X+ym3mH2L+/YPRyolPM1teNLjUayy9IyQ+t8+XSJB0KBBD3t46DozMdn+sgl/hmVj9ohHpH2LUXpuT7RmjTqa+jrIGXjDxQP9fmUCDINv7D3vxto6lU3NTC6vt+bPC3jFrMorTB1fbcqnyqF6wtCA0UvYYCgZcfZ3RnrpG+Ut7w51D/b3Y8tphFaZFValBN2ANSk3UIEED2e50gzrfPqW3dJuCCPh/eMXKSfRIpcAGEf3X//tg+U74BtyCnbxq1mBL1HVMcez+gPNDo1cRAy4SfZt9HcwtDgCBkwySh0lubMFIT80Cfd3dtnGCfRAZcf3Ht/rgABCKK540wpTUXYfmZRpggLvwRtLDkTfR2TffdL+0zFopq0sLbTbFGa4EAoi43YsC5MAlclxjhHxxvxD2nGhFS0jDMxhAoc5JiOsbnJv5NIaL6X8OjBqTFlOLsoWH8xJMpAgRoF2TKWORM1H/tewAbBWAJCDB+vxFopJFHjHAIUW/ndj0PB1SxK7MlJiQBgceTNqtGqmu8x/sHLYCWik7OzEKfGEwugPSYEWnk2siDMBdqBgJI/KMRWiD0AeR8wFQsLPHfh06RwMXnXCMNPzuTdjcyqEOBoCgFJxUfo5gocjzo2ejkT2kEBum5DgQxxGugsVcxZvyMiaGf53cZJcOZNc72mqa9XW7RakOB4CdYS6RCn3oBQQ+l8gM18bNMQMwu8V1VDrxmVNZwj5+ZtaoZNocCwSfwWjSRgICDG41WYhrBp41TjclW5VDp0eujCjpzndEFRnNNQHlb3aLVhgJBWUIGttXPyEYrMSBoJmOHYgsi3j7mkiIp9Davla9hxuea9P3qkz8BRC1Rg3jx5pq/tUQr3E+7yQxoCASPuBSzJbPhhaiZIz9jG7QBcf1dXSf7ZAez3nsGgURqzOq3jZTIqvVP5LNV5xH8bI+h3ZuNGlR70JBrQAgUb1RXzmyYNoAdVvOI2VsXmtQVPV8K3X3XAd+FRsrXaKFPppsJRqYzmiyyv2vckgtPITPerof+gRZLlE0TUGDySaNYZtFHFzfYPccYzb1mQcvOLQtNflA1KDltgmYmd/OK0b4d+Hx05n0UFq6eNsqFkoAPuSfT2iEQvEbwagTVQsfxG5RIwgHECz3D6DKjmOPn7WmYGt3AyV5kya+vtIZv/uWoeDRqqjJJmlTPhKDxCT3uKZkowMv4JSuhQiCEsx7bzoUqZNn0i0ayT7wYMACM1DrDWIIrjtCKbhir/gJN+ZPMDFX940d2z6+MwkUsNAbjwRX73otDGii7HhSzU3T2ViPWCrgY8NuNqHtT+hhE/96opjhTOe4h4aLK4oaOd+1CT6odmbpkzr6SQUUAv7b7p65JQLs8ZZTyH3ZYbnFYKvv4f7dhUghz/GpZn/eEqrLPO/o6d74thdZDzILeV11P2Kez3TNyMoup/CmBoCranxpTQ6MEVCE1gEOuUp1A+G5AjA/EkjGmTyEYflJLNjHHs5a1h2qqWBtaKzrLvszVR0yuEfBU3y6ppCEjO/GzWmaW0yyz0JraLbE5xbqLFvZKtRGf8jaWRlC486y9GV8CEHBdbzTXNLI3A+9YPyhZH8MkxICBZsARH8tfQP68q6gJxMxYQPA2HNX515mDAPkwq35gRKHNi0Y4xltbhTUWEEA0KESdsqA0dv3ipkQNJTU/2+/HAsLUAtiUqGHqfq7t/SEQlLpcFUPnWENsFi1dq4wacLBIh+8J1z7Wyf/Q0RAIZ6+49zUgWDFLO82tWg7r6CNtfir/uZiGdQlqj2l3AcIeM9T5ji5AWB8QCE+/azRW7oDM52FGvbK4YwKB3HnxQIb1yX2jWm7O/FVy3ztLOQQILG8eYbTLCDRyTZEzr5TBbG7TGgwVR9WZv4beKavbUlQ7aPVR5VGqT2itqm3o21bdWrUsPLDHtHGvUXVib4hGgFdfqJE9iGFgx7blcVT3Kg7IIO9C+Vr18v9QIKiiN1X6vi0DOEY/VnlcDvxiIjDZVeZ6KBBUO9e3kHMMAc/lHavSBpKHtHWpnnHn/qFAqNnxNJeBmprPUsHqFO1TtMpKcNFxHAIEv8ehCnVT9HQm75Ssxi5qKXW/WMauF9QCQcmPU+zB/TqP9Ej75KSOWOmWIopv2Pes4XtEYreu6uwX5wzOMffAwH7NiFoFLvr4QKYv2j1dszMJR4/jeQ422tXJKbTzjMdvu7HIVSHJhytO1BogqPQaFUPp9L+NCE8oj2azS7jjSSVqgEDHvSiikDerTTI1u6U6WW/EB/zfbYQ3Tqn/LR1XOkxETIZHCWS3pAc900orJ9RoD0lYGeW3JuYqquUnFPdo5oAg1MFMOGB+R1SqkfBUEYTGmQtseXvNiPQqnZgiqTIFalSRDYjDgfbyAPTvGvlqptZzk8S/dlXFtK6KgZigqd1j1buvc0AQimOI8x3PqTt1hHewWYPYdqzc+hSDnXun3/MYbi8rySN7flGm0dKGW8boTwWZVu2+TgFBNi21TU1hY2mjh4RHX0v3rnpgW9tTVXMsVNakSZ0g0xcIpbMc4amUNKpqOwYEb39SVbthqXdKqH4v5ao95taBLt0fbvHX/X5bXyqfUjUYCQZSJ88gW/adljYVV7UdA4JmewrdLWllj+iiw1IaiTV/nzoW0O/4JqqK+TxVg5HoX+q8JsxuafOKQFo8+S0Egh+4FLql7mvSyr4Tc88+KhfgJ4j/2+k2MKlQuO/hGGDD52sUfdVqg97OYunYfW82SgPLuzgHECEQRo6x93DNSmFnUO4zwjTSHyII5FDa89ASPoZ99KaHnAFtccR/Svv45wWEolkONULuoAwa8Idlyn+gsfAMYDQLYc1uo4OMOMqeS4kNOsff5xI6wrtyIC3H52pQJNe+RxX7w88BIFfNr75UtxsCwYcroU3HJHB0i7bLM6hc1COQOVTcrMILEi+EijHPl3eN+WNWEviUnz6m59jbll1P0qTFmZnogI/i0Ea17ePvMV7FTbs5HwHkfa9j7Gf2iU/wz27g+TMa4YdGu404g5l0M1vgucLfBvBC1D0lb3fKQe3zbq8NeZ5+8LffVIKiegEowlzf6AsntapgKBY1eK0ATwCCDR8ctOAZIi+gPLc/yiXmO3iTU/It+gzS1M8oy8ps9FpR7Ra9crtRqfqWY3P1ftn6llyMxqrKHKUSSnRcv0sQesKoOc4aCFUjzObOIOA5rjn5BfArB9H/4grmjhQ5x99o3aS0U3pIYYoc71qTAN84qGzerfElBtcjTD0T1/1+zcTc7yAgcOxwzQGYAlWLVpDP1XJkcXM7NauP6x6MdbYvk5dz8hRy16rt1ipjeGhZo5HmyeU1PifTBQh5mAkIuayobHHJNPiWUlXG+kkemWNAU/ypvqALzRXMPL8AIQ8EzfZUFlW5hdxScKqFWAitNRza6xNd9Q7LFyCUDQ9gIHzG+2bACaG5OEWWv91sVL1/IGiOd/stagpRfaRW5vCTO0o/Opp9zwKEWjF/ch/Oo66pSuximdo2LnvcvQChh9C28ZEFCNs4qj36tAChh9C28ZEFCNs4qj369D+S4+1njZS7yQAAAABJRU5ErkJggg==\" alt=\"dy/dx = f(x)/g(y)\" style=\"width: 65px; height: 36px;\" width=\"65\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21.5px; text-align: left; transform-origin: 384.5px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.9167px 7.66667px; transform-origin: 73.9167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith the initial condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 156.35px 7.66667px; transform-origin: 156.35px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.The test suite will ask for the value of the solution \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.8333px 7.66667px; transform-origin: 26.8333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.675px 7.66667px; transform-origin: 84.675px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Functions such as ode45, ode23, and ode15S are not allowed. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = separableODE(x,f,g,x0,y0)\r\n%  x  = point at which the value of the solution is requested\r\n%  f  = function of x\r\n%  g  = function of y\r\n%  x0 = point at which the initial condition is specified\r\n%  y0 = value of the solution at x = x0\r\n\r\ny = y0+(x-x0)*f(x0)/g(y0);\r\nend","test_suite":"%%\r\nf = @(x) x;\r\ng = @(y) y;\r\nx0 = 0; y0 = 4; x = 4; \r\ny_correct = sqrt(32);\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) cos(x);\r\ng = @(y) exp(y);\r\nx0 = 0; y0 = 1; x = 7*pi; \r\ny_correct = 1;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) ones(size(x));\r\ng = @(y) 1./y;\r\nx0 = 2; y0 = 3; x = 2.5; \r\ny_correct = 4.946163812100385;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) sinh(x);\r\ng = @(y) cosh(y);\r\nx0 = 0; y0 = 0.881373587019543; x = 5; \r\ny_correct = 5.000090791616095;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) exp(-x.^2);\r\ng = @(y) sqrt(y);\r\nx0 = 0; y0 = 4; x = 3; \r\ny_correct = 4.431659465773041;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) 1./(1+x.^2);\r\ng = @(y) log(y)+1;\r\nx0 = 0; y0 = 1; x = 1; \r\ny_correct = 1.622607687386726;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\r\n%%\r\nf = @(x) tan(x);\r\ng = @(y) sec(y);\r\nx0 = asec(-1-sqrt(2)); y0 = 3*pi/4; x = 2.302554350306210; \r\ny_correct = 7*pi/8;\r\ny = separableODE(x,f,g,x0,y0);\r\nassert(abs(y-y_correct)\u003c1e-12)\r\n\t\r\n%%\r\nfiletext = fileread('separableODE.m');\r\nnoODEfns  = ~contains(filetext, 'ode45') \u0026\u0026 ~contains(filetext, 'ode7') \u0026\u0026 ~contains(filetext, 'ode8') \u0026\u0026 ~contains(filetext, 'ode2') \u0026\u0026 ~contains(filetext, 'ode1');\r\nassert(noODEfns, 'No built-in ODE solvers allowed')\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":"2022-01-17T04:12:35.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-13T17:05:52.000Z","updated_at":"2022-01-17T04:12:35.000Z","published_at":"2021-03-13T17:09:14.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSolve the following ordinary differential equation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dy/dx = f(x)/g(y)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{dy}{dx}=\\\\frac{f(x)}{g(y)}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith the initial condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.The test suite will ask for the value of the solution \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Functions such as ode45, ode23, and ode15S are not allowed. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51387,"title":"Solve an ODE: second-order linear equation with constant coefficients","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 190.583px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 95.2917px; transform-origin: 407px 95.2917px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 314.317px 7.91667px; transform-origin: 314.317px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve a second-order linear ordinary differential equation with constant coefficients \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 7.91667px; transform-origin: 17.5px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.6667px; text-align: left; transform-origin: 384px 18.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQcAAABLCAYAAAB9RkIEAAAKYUlEQVR4nO2dzXHjOBSEOwdloOselIAicATKQBnovBenoPveVBuBcnAKjsHHvc4e6C42YYAAKYCizP6qWDM1lDX0I/DwfgHAGGOMMcYYY4wxZtvsAZwAXACcAeye+zjGmDVwAvAHwB3A1/ffvwC8PfOhjDHP5QjgE0NL4YpeQdiCMGajXBG3EGhB2HowZmOcv/9MTf47rBxasn/2AxhD3tFN+E90k/6W+TyVg92KOuzRuXBndO/i/bmPY8yQN3QT/g96yyFFiQIxZezRZYE+0MvfFplZFUf0g/Mw8rkTupiDTd+6XNDL3xaZWRUcnB9ID84dOtfjtNRDbQhmge7PfhBjQmjWXkc+c0Xe5TDzYLzn8uwHMUbZoTdpU1bBGY4ztOKAXv7HJz+LMQNYBfkH8VjCCXHFcMB4fMKUcYYLy8wKOKKb6B/f1xW9v/sR+fwJP6skgU6J3IN/P31/l6ZFYz+39SKqIzr34YJOPkwNqwJmepM9LTHO8h3GzEYHoTZP3dBbDWF+Xc3d2JWKT5yR/k5VDjFl9Juhov1ApxjPGKYwz/K5C3oFG1Okml2yu2dmc0A/0MKYgqbQQn+XK1PqGktnUhF9jTzPWPDzN7FDb52FvzPlFEsh79Ar0lgWwxkO8xBMP8ZWcaBXDrX9XY1jxIJsd2wnJUrrILbCUzl8Jn72HePxoDuc4TAz4eBL1S/E/N0aaAYkHLxcEbfgK+e6WKm4U1aUunYxZfoJB4XNDLQkOhX8Ky2ZnkNqVbxg3f0DDBgyuJpatUu+Z0z++8x9kup5OWJ7cZtXRAP1d6xkwySas7FsAzA0/VusPrGSYLo5ay6/PmCoWFMmfw5O6tTPq/zHBkvK+rhjuxmfV4CxtS/082uPbl4+tdJYzdGUyZpzOR4ltnKu3WpQcvIbQ3/3VEyAyjsXUIwpcVsN60azcuHCy3tP21FNB1RMQ+nK2DJroClNCqWm1VCSOZlDztfPMZYFAsqUB1H3g+4f06G1aCXHrcKFN6X4GWhutTCPooMzHESaItPBf0b9waEBT/pbNWnlFj3aJakpyvDnNYOkymNM9ppxOqN++rKle7k1VPGnxrt+ZnH3Qi0HXZno82jxE3eXvqG+FtNJFlZU1qDVoKbJP9d0H1MOV/STnXGEQ+b/YtzhE20yFFYO9eC7yslT58WihNYBdxj6Ql/cpBV2H2hjUqr70mLgtfjusTRsKVolSreNJee0ojjZD8hPeP2+FnUNVg710Hk3Bheg2Wn9HfqOSBa8cPegO8a7+Bi0UiVwlHtf6AZly/p8mk+tgpAtBrUqNJWXpqNypmDoOqiSBoaWW4klcJTPtqC1ctihkxm3JFQZ7jDsN0nJ9/R9b80BbV1YYtXBylh1bJbz939wkx/mTkyP+MNLwhfa6jlbDGo14Vn6/IXh2R0lgVxOCB4GpJYZFXxpUJHKvFVLd+uU9hc6eVEGDMhd0SsHHduxiaUKtVQOHH+PXqVy0VjCFOUwKRbHARqrXHyarzIRmsstI+AtBrUWHV0xjMWEFsFS0f0b2m6008o9o+UaPnvKpB7LsmnzYOlzqvv8yDVHOeTiVRqbKH63uYaalr5nLTgwWm9eUntQawqTgduQsUxQCy6J56hJbTmqYoi5AZzkMeuL1kPsd44dtDQGW98fvUqZ0i2r46hIOWgjVGxV0sG7puDRG/p4CAdGjZVOfdTYpVbU2FWKBv5SVs+SyoFt3o+ytBzpNkyZyEQrQkPOWHc3ryqH3HtTNyk7V7TYJRV04eCdI/SWqAn2iXomsA7cuVfO90v9f6mJry+19jvQvTzPqPeel5RjrGhrCrGAMFl7o5n+7lNiDtlFRjVmypctPXhmabhS5DIoUzlg3NzTQZj6TOlg0kjzmL84ZvY+igY8awZyl5Rjrs08h74HXSRZi7N25iiHUdmqQFIC0GCNd4HuqOlilRzso++ghUvBLMbSG83WlGOu4W/Kd6grM8dqWDpbAQw3Oiqpes1aZLmBOTVK/g+Af1d+/ZMTSgE1BzVXvFRRir6DnN/718LXo9SUY4n1lSN8F3ODsktnK4Bh3Cql5NUYyMZQckGuK35q5B3SSuI/1BFKy+u/nFAKqDmoOfFTgTfdiHdsRfwLy8vyUQWxNuUQxh3mpsSXzlYAw+rkVOww1mmbRLVNmN9lrl2bmJgRWLxpY2XUGtS5YDBfZmmZ+d8LX49SUzmoWR1TouwjGZNjWGm45rR9jFziYCyV+wMdnJ/oNOcb+lp8yP076qUKX51agzq1Y/YOP6v5fiM1lYOu+uGkZkdpyWpMS+5V962gpRkuNrrQFKMDlEpCFYAGMF5JMbA8tkWQrdag1jJp9pu8f//9ht+/41JN5QD0JdOcBFzQpihYrq5LB2drQjnc0G89yFLyyXAixV4S770KWu4aWkW1qD2ogfm+5ivTQo7AY7L8wOu5EymYSl5zjcai8OUycKp1HBZSmi0ppRRMQZpfyBviqSdmXH7LivAo2rnJfTW2LpuSgKV5Ens8/mJOie9gynbNPflLQldxTjvyb+GIfqywk3drMlgtB/QBuDva9m1QOWw9BRvS6uSxtRPumGXFsDLCcxpa+npcIbc0AUqgXF6hd6Am4bhzLGqlpHLTtWDl2NZ96hhM+71SiroWe1gprJol9om44HULWlqiXZGeJGZ1tN4ngoEmuxM/eWRjFGOao30btaE74dTUcOdw+ti6wYuyQ+eTM1AcuhzMcrTcwNdsHG1yqe3zskEsZi5vyYRmeTbTuDzIRtvwNYPDorELhmXzVALa1efsj6nGCf0ZDO8YRozHVvcDhmdo6GrFnntd4cY6R8+Jf/+NqHUQ/s5aXp6SfbjBjB6Cc0Tbc0fMRnhDN6j0EFY2gtDnTcEDR3Q7bWYddKdhNtqwr4LnPej1gWn7Or4y2tsfy9JQOYwFasN28iV29TYbgkU24UYlGikv7RQLW2lv6E3lMzrFk9ueayspu9yZm2OKQ1H3w6lgUw0qhthuOqocSrsltYFqiTMVXhWt/kvFXEpLptViswthqqDxhNjk5wCeUrarPvDc7bu2AK2CVMXplJJplbnlbR5GN0RNre4s251SMl1yvsbWKdkpfMoxAxrT2YpLZhqS2wU3Fywbo3QX5q2iZzfEXAp9N7nJrg1xW+y/MA3gYEpV3k3aAVdQV2UrWYepaIoylP0ew8NrKPuY2/eGfm+D2DFxNdrrzcbIHZKjLseUsl0ObA2QbamYqRTdn0EnL9O+4TEDsdOcwr0NQmXOdLHlbyahWYhYXEDNXg7Kku447nqdOg/R+fcOncisSzih3xxYt28/o7cOqDRu3/fCqkltb/YO5GYWmiYL4wIn/AxwcRUam9w8P4OEPjArL02HWg+0ErT4TN8PLTd1N2IT/565b0wRWjjDA1K5T6GubKy6CwcbYwuf6FcqdT/UB+aJXPZ/h/DEpZhFdsRPd47bxo3JMXffmCxHDFciXaVih+iEaEQ9NvGPmfvGmJXDcwNiUfNcjIHNPSn2mfvGGGOMMcYYY4wxxhhjjDHGGGOMMcYYY4zZEv8Da459lEyzJw8AAAAASUVORK5CYII=\" alt=\"ay\u0026quot;+by'+cy = 0\" style=\"width: 131.5px; height: 37.5px;\" width=\"131.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.625px; text-align: left; transform-origin: 384px 31.625px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.458px 7.91667px; transform-origin: 110.458px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith two of the three conditions: (1) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3833px 7.91667px; transform-origin: 14.3833px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, (2) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAoCAYAAAAYNNPaAAADoUlEQVR4nO2bbZWzMBCFrwccYKAGUFAFdVAHdYCFaqgEPGBhNdRC3x/hHrJsSPMxSWjfec7pnyVspnND5iMUUBRFURRFURTlq+kAnCLGnwCMhWxx0S/zdRXnPDKxelXnuXz6gLEnABPqizsAmBvMe0Ri9KrOGcALwE/A2NMyrpWoF5hF9T8To1cT7jAGXt+M62C+xLm4RX7uqBtuj0aoXs14ImzXucOEutZ0MDYPrQ1pRKheTTghbLX3y7ijiDjiwFt+QUL1agaFCdmdjpS7cIG3Dr+1CdWrGTPei9LBiHcrb04UM4BHayMqE6JXNh1MKDrDiO4qJXnN7l30CNt1LogLd5013/aeDma7llic42LXp1FaryyGZeIJxrkvuCsgXttulyF9DFYV7+hhdowJJnHczney/v6CWag5xC50G/ot9xPbB6qhlxg08om/MfbiuRbyf5+R99ywOmXA2nLg7jQh3zkD0hNU276cT063upReYrDhtff0U8hYXgn3sRqh4BPkKxPOkRI+pXaoHLFL6SUKQ8rdce2ONFFTFpRty3PHHgle+OzEvIReojDf2cZehpyUMJO6oB5Yn0DfvB2M3fxMCA8lqbYdBWm9cnzphLF3m6zekH5ckSra1bJlLzSw6/2wxvC+EEd8+oKS1CvXl07Y8LOT1ZzdCUgXjU+fr6LjmG2l9sT7EpnftWUOlVtYSOqV40svP/idW4zIO0ydEwy6LHb4cgQ2TF07GCsgX0uAVV7KgjpClUck9Mr1pReu1Blr7yenGmEuFApfcRmse13nblwQrsVKwX2OZZXUsg8lUdJL6JXrSy92XJ6R30CMicPdZk47j+IW/sDaCX7nBF+o5ZjDnmsFIqFXri+92D0giYSVcX7vi45Y+0wzfj8Jti0j1uoDCHOCL9SmhOIjIqFXri/fwv6P1HvHvoNYO6dwjfnZuZ7jBOYMubvvUcjVq8qCkmyKcVt2VR7MR/bE7XeuMxymOOGKxscSwuTqlePLoH9eons8Q/YVlpxEkueC34CEXsWS8gHlXsBirJc88WYo3MJS1/Xuzw3fkTsBsnql+NILy/WSv9eS/tWJrxnncjTL6kP+fCgSab1iffkHvmA2YW0k1njn+wa5w14eF9jVDfOB7RPFlsShf+DoobReMb50Yh++lt6Ztlwgt6h6rAeZI8z3cr3p+cDnLiagjl4hvtyFr9O2Kp1rVlhd5flK0FovRVEURVEURVEURVGUEP4BkX7JQnEwa3oAAAAASUVORK5CYII=\" alt=\"y'(x0) = y'0\" style=\"width: 74px; height: 20px;\" width=\"74\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and (3) \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 126.783px 7.91667px; transform-origin: 126.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Two of the three elements of the vector bc will be assigned numerical values, and the third will be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.95px 7.91667px; transform-origin: 50.95px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 374.975px 7.91667px; transform-origin: 374.975px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEquations of this form appear in many applications, such as spring-mass-damper systems, RLC circuits, small-amplitude oscillations of a pendulum, and steady advection, dispersion, and decay of a contaminant in a river or groundwater. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc)\r\n%  x     = values of the independent variable at which the dependent variable y is requested\r\n%  coeff = coefficients in the ODE [a b c]\r\n%  bc    = boundary conditions [y0 yp0 y1] = [y(x0) y'(x0) y(x1)]\r\n%  Xbc   = values of x at which the boundary conditions are specified [x0 x1]\r\n\r\n   y = f(x,coeff,bc,Xbc);\r\nend","test_suite":"%%\r\nx = (0:0.25:2)*pi;\r\ncoeff = [1 0 1];\r\nbc = [0 1 NaN];\r\nXbc = [0 NaN];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\na = 1/sqrt(2);\r\ny_correct = [0 a 1 a 0 -a -1 -a 0]; \r\nassert(all(abs(y-y_correct)\u003c1e-15))\r\n\r\n%%\r\nx = [0 0.1 2.3 4.56];\r\ncoeff = [2 7 -15];\r\nbc = [3 0 NaN];\r\nXbc = [0 NaN];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [3 3.101061786097092 72.69322003332921 2156.513387836726]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [1/7 1/2 5/6];\r\ncoeff = [9 24 16];\r\nbc = [1 NaN 0];\r\nXbc = [0 1];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [0.7084846608207754 0.256708559516296 0.05486549796798426]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [1.2 3.4 4.7 5];\r\ncoeff = [1 2 3];\r\nbc = [NaN 2 3];\r\nXbc = [1 5];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [394.630646389682 -43.24486540256236 -1.2288043607505912 3]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [0.001 0.01 0.015 0.1 0.5 0.7512];\r\ncoeff = [1 100 1];\r\nbc = [2 NaN 1];\r\nXbc = [0 1];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [1.9057927709196938 1.374168410483207 1.2308202441263283 1.0090865186701083 1.005013023466313 1.002491347110156]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nx = [-1.42 -0.56 0 1.8 2.78 4];\r\ncoeff = [9 12 4];\r\nbc = [NaN pi exp(1)];\r\nXbc = [-2 4];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = [25.647244651987194 21.17888752808947 (7*exp(5)-12*pi)/(15*exp(4/3)) 8.21698074712973 5.101899094149195 exp(1)]; \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))\r\n\r\n%%\r\nA = 3; x1 = 5; b = 2;\r\nx = x1*rand(1,4);\r\ncoeff = [1 0 -b^2];\r\nbc = [A NaN 0];\r\nXbc = [0 x1];\r\ny = secondOrderLinearConstCoeffODE(x,coeff,bc,Xbc);\r\ny_correct = A*sinh(b*(x1-x))/sinh(b*x1); \r\nassert(all(abs((y-y_correct)./y_correct)\u003c1e-12))","published":true,"deleted":false,"likes_count":3,"comments_count":1,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-10T17:15:55.000Z","updated_at":"2021-04-10T17:23:45.000Z","published_at":"2021-04-10T17:23:45.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve a second-order linear ordinary differential equation with constant coefficients \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"b\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"c\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"ay\u0026quot;+by'+cy = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea \\\\frac{d^2y}{dx^2}+b\\\\frac{dy}{dx}+cy=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith two of the three conditions: (1) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, (2) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(x0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(x_0) = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and (3) \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Two of the three elements of the vector bc will be assigned numerical values, and the third will be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEquations of this form appear in many applications, such as spring-mass-damper systems, RLC circuits, small-amplitude oscillations of a pendulum, and steady advection, dispersion, and decay of a contaminant in a river or groundwater. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51087,"title":"Solve an ODE: equation for a 2D laminar jet","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f’’’ + 2(f’2+ff”) = 0\" style=\"width: 147.5px; height: 20px;\" width=\"147.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOYAAAAmCAYAAADQgucPAAAFwElEQVR4nO2c4ZHiMAyFXw/pgAZogAq2Ajqgg+2AFqiBEuiBFlIDLXA/4jfRMkSSHYU4nL+ZzM0tIYntJ1mSTYBGo9FoNBqNRqPRaDQajUbDwQ7AIfM7HYBL+jfqGc6B19sS+3RY7NKhkTuOZ+e9vVxgP2PDYA+gB3DD0KEPACfH97r0HWtAdwCOAH7TYZ1/AHDH/2OcRwx9fsHQnz2mDWsP4Imhf6a4pnN+nPe/pGfQ6NL1OIbWtXfpGSONfU1yNTybA4ZBvKb/n9L/H9ANo8PQ8Zpn5mz6TNc9YBThFXrjjtDF9y38YuwfYDSq28T55/T5ZeLzLn3+hG/GuqRnsJ6RjuOAwSjv6dAMlA5/yzPnHA3PuukjHey8H+jCIFcMItGufU/XevXGdAY99IZdjHtsHfaDdEA0vClj6dPnU/12xF9Hq3GC7fwoyldHQO1YM/OWHWyEhougt/YMooQeQ5tRKbCpQeGAa/fm4OfmS1vhhr+zpQXDWK3POON60osn9L6lk37i/VjTCfQTn5Mb7Fm5RiI0XAQ9npVfvNJD72gOuub55aBrHvec7vdt0Mi8IScwphma0THEsmA+q8HZeeo8GTZrkQ3HekshbaSGs6C3y+0wfs8zW2oeWQ7qVL4EjB0U1vBKoLfNcTp36EZHoXhnS61Ppeg0J8xQz2pHj22lJZEazoIhT+5sdIff01oGzPMejnuGhwsrw2jFO6AUgeUQPeKn6DToOCwDludpDuEMe5xrIlrDKieM5V4Koxd/OxkPQU9r5QtskPWwN/gH1RJS7XQY+1l645v4u5ZSHGEbnZXryfMs5+odm19xnpYrcwbeSr0gWsMqFID0clf4hAGMYazWuaxW5TbKEmXpoB4wtm/OMTc/2olryXafxd+t9lnP4HnGXOeaY5ielKSkCCT7bs7h1c8SGnYhOzNH7PyeNlCyUVaYfBXnat724DhnCtnWOUdkSZyDaVW2l8Dbl7LtmsGfxHlWulFawZSamnN49bOEhl3wYrnCoKA0ZKOsQfCGQaxglnjbqBkz0oCYRljh5BKwzy2H7J0xcsb7gbI2R82YXue6hIZdlAoj1zC9i9deD/4NBSC5TLLG2l6JYWozpqzuW+NzwzYKQEtpWEWuz+SWrz2GKYUXGZ+vNcNEI4W8xhKQ1zDpvKNyTMCnnxpYSsMqcxZFvR1b0igrzPgWw5QV2TU26XsNM7oqK6+5BZbQsMocYXg7lovOVg7rPW9ORa+Wqixhm9faP+o1TM8Ce855gG+Z5h2frsoC8Ro2oXGVCIODYD2AHCzNi/Aca7AY85cYZm1V2dI0IgpGTFa0JCMrbSbMqTCXRj2frsoC8Ro28eYD72Bp3LPeZhU4ckLqOYvTNVVlpcBmr3kVklPhjtorK89dcx0zx7l+dK+sTGpLhMHvezyPd2e+x9Nw1tv6D6flmt+aG7q9G90pvKnK7FF8bo0NndKW9jxHalglQhg9fLOt9ls2GrhViif8Ue7WKd2fHM0l4xkovtcx7zDOqB5j86ZBNRGpYRV28hxh5GxG5qtHOMvKX397f9nOEGit0C8SCrkkjYjE+ysUIt+aIN9g8IB/Brxj/XaXEKFhE1r/nMIDY+8cQ9lj3EB/Ql6sf8I6W9eikflYDeGcN/Ihr++8ycn3ObtsZQP7O+ZoWEUKY+5Fz/jcmmKPgK1OFSDztRr4wecc3gXfsWtrEZioR+RqzC+WDi9/8R25JTCmETX9WPiK5cPLA4LysG+Ab/aSMw0LD1HhxNJvQNtjuwO6x9D/MmTlS89qCslZ3FgqtKYDryF0rwK5qN5hjPGjveMPhpA2WmwUzFbfSfq6iYPV8BoLWEu+//WKOtu8GpxtGG7mJvq594o0zg4Lvq/zQzBtuGEwytrzZO+Lu3M4oxnlW3YYxHDC8uFg5IzZBV9vLfYYIpcjttOeyOfcYgrSaDQajUaj0Wg0Go1Go9Fo+PgHd/0S1qi19d4AAAAASUVORK5CYII=\" alt=\"f(0) = f''(0) = 0\" style=\"width: 115px; height: 19px;\" width=\"115\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the constraint         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral of f'^2 from -infinity to infinity = a\" style=\"width: 119.5px; height: 44px;\" width=\"119.5\" height=\"44\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves a jet in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x, y\" style=\"width: 24.5px; height: 18px;\" width=\"24.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plane emanating from a source at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(x,y) = (0,0)\" style=\"width: 87.5px; height: 19px;\" width=\"87.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. It can be solved by expressing the velocities \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the problem above, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(eta)\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is proportional to the streamfunction. The conditions at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"eta = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e define the line \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = 0\" style=\"width: 37px; height: 18px;\" width=\"37\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a line of symmetry; the transverse velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the shear stress are zero there. The condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e states that the streamwise velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAAmCAYAAADZRYUwAAACr0lEQVRoge2afZGzMBDGHw84iIEaQEEV1AEOcIAFNCChHmqhGmrh3j+SnWxLCNlN3rbc7W+GuWk7JCQP+5kDDMMwDMM4GgOA86cfwkjzAHD59EP8NjoAI4C+YowzvDg5HLx4I7yVnSrm+zNcAfzAb5iWOVwpOjbH67WE340EA+JGacXpwv2peNMBuAG4w1tNH/5ysRblvL+aE57fYq04F/jNTzEDmJC2jiXM+4B3eUagg9/QEfXiLPACvOLgLWSLM5vb4g9jht84cklacRy2N9chbxE9ouVY3AlQZkUbVyPOAB9TNFC8S1mdmFMYkKedffh8FLN0WNcjNeLclPeRWyXrVXMKg9zDg/TwbuGGuLDSgObgxay9tDXJFeuUVysOJRTSYO4QM7iqROASHuCOtcI/7LdSyM/WXpq3ddxZh3TMCbI02IV7HnheyyicF0BeGLDBJT6zleVI3Si95an7tOJQ7VICX/eCSoF4mpdyIW7n92+Cp80pNOJQUqGNFx38S01zF9c6HaKyW2ZLVnWENJDS5i004uTaNRKoEN3qMKzghdmW+6D2Q27R3wB5gAnbLpK3UUoSDnp5WxwP8C5F0ctBVrO18TyoS4PZu7M1vvmtEo5cu0YDzbkbv0o2njftpIH53dka1WG5i8a8su9y69pq12gpLkf4w6bMdkC0LE28+VS2lkMieK5do4HcWlF4yIlD5kxBjAacoczVvwSJOAPkdd2eFRYXsjxAUWuBTgqp3iHLoYzl6OcREnEk7Rp+VrRgLcAMRWLB0zs+OLmwx8v3R6dUHHJppS2XVHy9IrbCJsFYT5wRz7pfBzhas3OP0jg2QV46OMRm8Yh4Gmo0RtKuMd4Iuahv74b8SVq1a4z/QKt2jdGYC/b/YdD4EK3bNUZDWrZrjMaYMIZhGG34B3egUSmJW27bAAAAAElFTkSuQmCC\" alt=\"a = 4/3\" style=\"width: 51.5px; height: 19px;\" width=\"51.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = lamjetODE(eta,a)\r\n  f = g(eta,a);\r\nend","test_suite":"%%\r\na = 4/3; \r\neta = 0:0.2:1;\r\nf_correct = [0 0.197375320224904 0.379948962255225 0.537049566998035 0.664036770267849 0.761594155955765];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 4/3; \r\neta = log(2);\r\nf_correct = 0.6;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\neta = pi;\r\nf_correct = 0.902552583791843;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.5;\r\neta = 1.5;\r\nf_correct = 0.572446107496431;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.75;\r\neta = -4:0;\r\nf_correct = [-0.823247562011528 -0.813902901498664 -0.766864130068021 -0.559711742572462 0];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = rand;\r\neta = 4*rand;\r\nassert(abs(lamjetODE(eta,a)+lamjetODE(-eta,a))\u003c1e-10)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2021-03-21T03:15:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-21T02:07:31.000Z","updated_at":"2021-03-22T00:42:37.000Z","published_at":"2021-03-21T02:24:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f’’’ + 2(f’2+ff”) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime\\\\prime + 2(f\\\\prime^2+ff\\\\prime\\\\prime) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f''(0) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f\\\\prime\\\\prime(0) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the constraint         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral of f'^2 from -infinity to infinity = a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty\\\\left[f\\\\prime(\\\\eta)\\\\right]^2d\\\\eta=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves a jet in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x, y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex, y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plane emanating from a source at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(x,y) = (0,0)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x,y) = (0,0)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It can be solved by expressing the velocities \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In the problem above, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(eta)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\eta)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is proportional to the streamfunction. The conditions at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e define the line \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a line of symmetry; the transverse velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the shear stress are zero there. The condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e states that the streamwise velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 4/3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 4/3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51808,"title":"Determine when snow started","description":"R.P. Agnew posed the following problem: It starts snowing in the morning and continues steadily throughout the day. A snowplow that removes snow at a constant rate starts plowing at noon. It plows 2 miles in the first hour and 1 mile in the second.  What time did it start snowing?\r\nWrite a function to solve this problem. The inputs will be the time the snowplow started (given as a string), two consecutive time intervals  and , and the distances  and  traveled in the first and second time intervals, respectively. Return the time that the snow started, rounded to the nearest minute, as a string.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 135.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 67.625px; transform-origin: 407px 67.625px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 367.333px 7.91667px; transform-origin: 367.333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eR.P. Agnew posed the following problem: It starts snowing in the morning and continues steadily throughout the day. A snowplow that removes snow at a constant rate starts plowing at noon. It plows 2 miles in the first hour and 1 mile in the second.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.2917px 7.91667px; transform-origin: 95.2917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWhat time did it start snowing?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.625px; text-align: left; transform-origin: 384px 31.625px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.275px 7.91667px; transform-origin: 380.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem. The inputs will be the time the snowplow started (given as a string), two consecutive time intervals \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dt1\" style=\"width: 23px; height: 20px;\" width=\"23\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dt2\" style=\"width: 23px; height: 20px;\" width=\"23\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.675px 7.91667px; transform-origin: 60.675px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and the distances \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dx1\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"dx2\" style=\"width: 25.5px; height: 20px;\" width=\"25.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 184.625px 7.91667px; transform-origin: 184.625px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e traveled in the first and second time intervals, respectively. Return the time that the snow started, rounded to the nearest minute, as a string.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tstr = snowplow(t0str,dx1,dx2,dt1,dt2)\r\n%  tstr = string denoting the time the snowplow started\r\n%  dt1  = first time interval\r\n%  dt2  = second time interval (immediately follows the first)\r\n%  dx1  = distance plowed during the first interval\r\n%  dx2  = distance plowed during the second interval\r\n\r\n  tstr = datestr((dt2-dt1)*dx2/dx1) + t0str;\r\n","test_suite":"%% Original problem\r\nt0str = '12:00';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 2; % miles\r\ndx2 = 1; % mile\r\ntstr_correct = '11:23';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '12:00';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 3.22; % km\r\ndx2 = 1.61; % km\r\ntstr_correct = '11:23';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '12:00';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 3.11; % km\r\ndx2 = 1.73; % km\r\ntstr_correct = '11:09';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '12:00';\r\ndt1 = 2; % hour\r\ndt2 = 1; % hour\r\ndx1 = 4.24; % km\r\ndx2 = 1.26; % km\r\ntstr_correct = '10:29';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '13:30';\r\ndt1 = 1; % hour\r\ndt2 = 1; % hour\r\ndx1 = 3.11; % km\r\ndx2 = 1.73; % km\r\ntstr_correct = '12:39';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))\r\n\r\n%% \r\nt0str = '09:53';\r\ndt1 = 0.75; % hour\r\ndt2 = 1; % hour\r\ndx1 = 2.28; % km\r\ndx2 = 1.99; % km\r\ntstr_correct = '08:35';\r\nassert(isequal(snowplow(t0str,dx1,dx2,dt1,dt2),tstr_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-22T22:48:59.000Z","updated_at":"2026-01-02T12:00:02.000Z","published_at":"2021-05-22T22:52:55.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eR.P. Agnew posed the following problem: It starts snowing in the morning and continues steadily throughout the day. A snowplow that removes snow at a constant rate starts plowing at noon. It plows 2 miles in the first hour and 1 mile in the second.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eWhat time did it start snowing?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem. The inputs will be the time the snowplow started (given as a string), two consecutive time intervals \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dt1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta t_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dt2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta t_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the distances \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dx1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"dx2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta x_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e traveled in the first and second time intervals, respectively. Return the time that the snow started, rounded to the nearest minute, as a string.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":53995,"title":"Solve an ODE: nonlinear third-order equation","description":"Write a function to solve the ordinary differential equation\r\n\r\non the domain  with , , and either  or . The input variable ord indicates the order (either 1 or 2) of the specified derivative. The function should return the values of  at the requested values of the independent variable .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 124px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 62px; transform-origin: 407.5px 62px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 176.35px 7.66667px; transform-origin: 176.35px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the ordinary differential equation\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'y'' = y'''\" style=\"width: 71px; height: 18px;\" width=\"71\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 64px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 32px; text-align: left; transform-origin: 384.5px 32px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.2917px 7.66667px; transform-origin: 46.2917px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eon the domain \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH0AAAAkCAYAAACkATUaAAAEKklEQVR4Xu1aOc9OQRT+vh8g1kolaCSEBNEgoUAoSRCNRGJNFIhdZ4keoVCKJWohCgWNECEKhaWR6Cz/gOeRmeTkmrlz75yZu3jnJiff+71z5yzPM3PmzMw7PVWeiUNgeuIiLgFPFdIncBAU0gvpE4jABIZcZnoh3YvAQrTMN60vJhCnsYQ8F47OhHypczg003eh80XIa8gbyAIIvzsDeQj5PhY0/nM/SfY+yGnDza1Y0m+g4yHIZcg5oeQAPt+EPIHs6YH4rbD5sge7ucaNNp6ThuzZxsGD+BtFuiX2MxSscQD8GN9tdgyIXMBQr806i/B5BeRdTmMd6NbGwyWXs/s9ZL2ZoHQ7inSmio8QjpzqLLdY0OG75p/cBEhwaPI+5Cykdt3qgLRYEzniWQtnnmtmup3l1LEN8sgRHUcZswAf38CIBUUOLNYTnNmW7Gv4MNZC0kV2qnjUpL8CsKsM0ATcN6N+oI3ZgOQv1jIs+ucEJ6GbjVV1EY+a9N8mnJ/4O6cmNLuu85W6wdEEHVt97u9oZtPeEsg3x6Beju9mKDNK1/GoSGfAbyNI9y0DIcLlVsNWn1yzU6U9aZ+2jkE2iUwmdyBsvwNhgconJqYu45GxqUiXndvM9GDFWGG/D3DkwYXdjtItW4hyWeNSNcsQ36ZA7SOeLKRzxnE98j3cH141jW1IZ6F4BWJnNmcbC7YuCzQ5uHfD9jLjz+GaeH1NQ4snyEX1RE6CQTK21IBwCW3cOvEJGhJ6WPmfgPDgxz7cAdyGdLkNs7ULTxv5rI4gnF2GEM8o0jvBcqXELsmXhWibVO4bG33GoyKdAcVU7xrQ+gJLLk/zEHeqe4Q+4lGT/gkAcAvWtJDje9zXa1OzD6wH0J3jyFUeQq2DjdQ1RZfxqEl3VbaulCbXxNj10KXXtcdlfXEqIflya0ofcp0qymVMnkGkjkdNerWyvedgRoLWpohrWytVT7NSgMVBxVu685C9EO7LWcylHLi+OHPEQ1tq0qnEFjm+Ct5W7r5buLbkht6vgqWpITiI6Tevi13rOtP+V4jrziHkZ9P2lPEkI53bEI5+7qWrAMtbuBxrYR1wFqztLVI976uXQjiASShjs1vRalaj7eMdzXraionHhU+SmU7FTOHPjIUNBmQCxkMbFm4xx5RNZ0LoPfrRtHC0tQd1ujKTvTjytYd8SdHeJp6qPU7C65CdpoH8HIF4dyOhn0tR4Q4ISefx5C8zEJ62AD0FKBodTOEbjb8XHGAwExytadfYzt3Xxla1Q5748zYeeP1Dfoj0HE7bWyyN7g91I1mjOKLv6OLpg3R5EhaB8d8umkIu1qav3+ji6YN0Vv4rlcjzYqTpmq40Few+unj6ID2IYnkhLwKF9Lz4DlJ7IX2QtOR1qpCeF99Bai+kD5KWvE4V0vPiO0jtfwC9TyI0o8V1UAAAAABJRU5ErkJggg==\" alt=\"0 \u003c= x \u003c= 1\" style=\"width: 62.5px; height: 18px;\" width=\"62.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.3333px 7.66667px; transform-origin: 16.3333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(0) = y0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(1) = y1\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.95px 7.66667px; transform-origin: 36.95px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and either \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'(0) = y'0\" style=\"width: 70px; height: 20px;\" width=\"70\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.66667px; transform-origin: 10.1083px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAoCAYAAADkIfgQAAAG1klEQVR4Xu1bOasmRRSd+QWDOpEYPFwCYUAZt0QDDRwXEMHAERMDcQ0V18BAHbfRyBUMXiDMCIIgKGqg4BK4gYJgoDOIgZEL4g/Qc6QP1Fev6tbt7qr+5vNVw6V576u+de+tU3er7r17+tUtsLAF9i48X5+uW2BPB10HweIW6KBb3OR9wg66joHFLbCJoHsZVvoEdLyStZ4En79Az1bi19kULNAadO9j/nNBl4F+T8hyHf73Buhh0GuO1RoDuMsHfr/ifrLA+wH8vgW6xyHD/3kI7fAM6ELQdxlFS2tatE9L0F2A2b8F/Qk6LwM6equbQQ86PA0BR16PGlqdjd/uBx0GPT2MuwP3n4Y5cobkUPL/2SFH0agbPOAPyH4a6ArQZwk9PGtaVL8l6Bi2HgEdyQDlDPz/twFIF+NueSOC6AlQzmNSURrkYxANdxNIAOM8X4DOAV0Pei9jFY77EXRDxuBFY274AEadd0Ffgy7J6FJaU5cJWoKO3uV0UA5QBNIxA5RSQGBg6MvlcSGw7sK4OFQz1H46MNyPeyrU82cPuF2G3cBB9PR3FzZmaU1darcCHcPciQKgqCQXueTlmGcwRDI3zF3KRRh+c/xoMHq7N4d5c7zoKRmad1thQfvQfjkv51nTtYLuTsz+VAFQXNxXQFaOJi9XKjSUi1ihQTuZhrG8HUMI80wL5C7jbtAg5WpW+uFZU5fKrTwdK5y3QbmKlOFuG2TlaFRAIdiqppSLcDxBnKtAabRXB6vcgnsuVCsUWwvgMu4GDWKkODjYOyd2aU3d6npAR29zPugHUJwL0eWeGf2m/MoCFL3JLwYopQCBcTWIuWHuUmjl76l8Ts+FeZ0FTo7/B5QrgNzGXeNAei5eqWqddvg7+u0r/H17Zjz5eNbUrW4OdBTsWhATS5bQuQVVWPsAA24FEZQUkIDLVYnkRf6e/hn5fwm6xtBIbZecjHo0BJ0Vhjmei0BdrHlzIqk/6F6ExMAYFB5e3HxXgQ4Ng5lTxxs/9PahJ2dEsZrtnjX1yPjfmJKnC6vClBLePMktUDBQLZWSV6Lbl6G9no4Js+U9xbNkn5Re9JJzr1KxY/EPgRXbQ20lOpK1eXKPUZVXUdFU05DAY8WTq3qmLoA8U6lxHILOyv0ohwBRAp02k8c+sX6UZ+71ERjMqZ5VqTMCxd5aVejaclaPUeVxaMjU7mh1djkFdLlOOmUP9SiBTnmixW8usFo+ryYu54grddnBs/ZNZPROrLyJuRB3TlhQcFeFJwC1BJ0CulrhddNBpxYI1yKu1Bm5bgTxvpbLC7owTwhDWEsFvKCbUr2WcibPwfdaFmzEpCryYl1bOQl60McD+RjGmRrtqKC9oMvtnFYKUHYv6MKc0+vpSkm0PLvXPiEW1lW9xnhMRadWTkJN/A8hxL1DJJSj2pFnjzGqdo6qSTK9srGbZuJfql6VGNPolgcLPWIpiWYxcCnIqnBzTmfd1avkCqMTj//4ziBffGiRCqnwinNgYoZdj5Uicwzo4p3DNzJK56YjokFyKD0pqdQvUwVLBXPHV5I/1fqJJ/fOmxL6VKheKVcYnbjJDoC2QLXfGQwLtLho0bqsgHEM6LRzWPnRjXLxrHPTuYDj897WRdj4TVWcoWGsEMw5NbbUqqmhX2seYXRio986c54qi2yfargruqykM2NAFy6s5VGmCp96Tueqpf4bn1WbIBVipXyqbxXP6znvraljS15hD9M6b54jg2xrgW7F7mNAR8GUr5RyojlKxM963kbRM/KM3FkvgNjakYcOj+os+Vo1u2vaxMvLAoSXR2mcB3QrgJwCulJiXxJy7O9U6iGQN6nXufFFw0Tf4P45yDoLlkwKra28wljd544XIFhIlL4TmTpXU9DpdaSl3zPT+e9zsIrn452pxuNzOjQvFS5z5ljyWYbXbVCtL+dSsiuSWOF1kqdjW4IPtq5WcwtCwL8Dyn3gU2MhdRjONpD1AU+NuZbgwRyXB/u1q9VY9iaFBD0Nd8xjzhDVyqBM8G8DtfBC0vH5xl6hlW1ivrTVfaDaL2Hk5FeuH6drKmRWaoBUTqfF5ZsOb4FYDb4Oah3aPAvC8LfVYPeyeKj5AbdHl1pjuGFeGpi9iPtZoNKXc7XmFh+rOcxCcOW9vhTowjKbTE+1pJqbYl/FTUAgfw/yFBq1F6sGv7CVRX6plzJqzGPx0DFY+MKtcr0dnY4U6Jjb8EMYIvQoqFXV09oQu4m/CqBtKN2yaLBsqg/deWfHgBUzPe+Oj7bHtkx200J2XRtZoIOukWE727wFOug6Oha3QAfd4ibvE3bQdQwsboEOusVN3ifsoOsYWNwC/wJfybU4YZYqZwAAAABJRU5ErkJggg==\" alt=\"y''(0) = y''0\" style=\"width: 78.5px; height: 20px;\" width=\"78.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 7.66667px; transform-origin: 61.4583px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The input variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.66667px; transform-origin: 11.55px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; \"\u003eord\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.5083px 7.66667px; transform-origin: 31.5083px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e indicates the order (either 1 or 2) of the specified derivative. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x)\" style=\"width: 30px; height: 18.5px;\" width=\"30\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 7.66667px; transform-origin: 84.4px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the requested values of the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solve3ordNLODE(x,y0,y1,yd,ord)\r\n%  x = requested values of the independent variable\r\n%  y = values of function at the requested values of x\r\n%  y0, y1 = values of the function at x = 0 and x = 1\r\n%  yd = value of either y'(0) or y''(0)\r\n%  ord = 1 if yd is the first derivative or 2 if it is the second derivative\r\n\r\n   y = A1*exp(r1*x)+A2*exp(r2*x)+A3*exp(r3*x);\r\nend","test_suite":"%%\r\ny0 = 2.231252941;\r\ny1 = 6.297567308;\r\nyd = 1.557407725;\r\nord = 1;\r\nx  = [0.3 0.5 0.8];\r\ny_correct = [2.790588219 3.308319582 4.544300275];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 2.231252941;\r\ny1 = 6.297567308;\r\nyd = 1.71275941;\r\nord = 2;\r\nx  = [0.3 0.5 0.8];\r\ny_correct = [2.790588219 3.308319582 4.544300275];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 3.287682072;\r\ny1 = 9.107569130;\r\nyd = 1.154700538;\r\nord = 1;\r\nx  = 0.25:0.25:0.75;\r\ny_correct = [3.669824691 4.306714708 5.456245713];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 3.287682072;\r\ny1 = 9.107569130;\r\nyd = 8/3;\r\nord = 2;\r\nx  = 0.25:0.25:0.75;\r\ny_correct = [3.669824691 4.306714708 5.456245713];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 3.693147181;\r\ny1 = 3.046411846;\r\nyd = -2;\r\nord = 1;\r\nx  = 0.2:0.2:0.8;\r\ny_correct = [3.36426198 3.152361229 3.034571214 3.000213221];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))\r\n\r\n%%\r\ny0 = 0.722781494;\r\ny1 = 2.637280183;\r\nyd = 1.030077779;\r\nord = 2;\r\nx  = 0.2:0.2:0.8;\r\ny_correct = [0.950884887 1.231252941 1.581096155 2.030246566];\r\nassert(all(abs(solve3ordNLODE(x,y0,y1,yd,ord)-y_correct)\u003c1e-6))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-02-07T03:08:13.000Z","updated_at":"2022-02-07T13:11:31.000Z","published_at":"2022-02-07T03:11:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the ordinary differential equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'y'' = y'''\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime y\\\\prime\\\\prime = y\\\\prime\\\\prime\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eon the domain \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"0 \u0026lt;= x \u0026lt;= 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \\\\le x \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and either \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(0) = y\\\\prime\\n_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y''(0) = y''0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime\\\\prime(0) = y\\\\prime\\n\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The input variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eord\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e indicates the order (either 1 or 2) of the specified derivative. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the requested values of the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51783,"title":"Solve an ODE: equation B","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 213.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 106.625px; transform-origin: 407px 106.625px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.358px 7.91667px; transform-origin: 211.358px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve the following ordinary differential equation:  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 39px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.5px; text-align: left; transform-origin: 384px 19.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\" style=\"width: 181.5px; height: 39px;\" width=\"181.5\" height=\"39\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.3917px 7.91667px; transform-origin: 14.3917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x1) = y1\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 35.0083px 7.91667px; transform-origin: 35.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and either \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y(x0) = y0\" style=\"width: 64.5px; height: 20px;\" width=\"64.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.1083px 7.91667px; transform-origin: 10.1083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y'(x0) = y'0\" style=\"width: 74px; height: 20px;\" width=\"74\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.1167px 7.91667px; transform-origin: 38.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Along with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25.275px 7.91667px; transform-origin: 25.275px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, one of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.91667px; transform-origin: 7.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ey0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eyp0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.708px 7.91667px; transform-origin: 109.708px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e will be assigned numerical values, and the other will be \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 7.91667px; transform-origin: 11.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eNaN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.742px 7.91667px; transform-origin: 128.742px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function should return the values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 80.9px 7.91667px; transform-origin: 80.9px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at the specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.483px 7.91667px; transform-origin: 194.483px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne of the applications for this equation is in groundwater. For \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 1\" style=\"width: 36.5px; height: 18px;\" width=\"36.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"p = 0\" style=\"width: 38px; height: 18px;\" width=\"38\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 136.567px 7.91667px; transform-origin: 136.567px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 141.975px 7.91667px; transform-origin: 141.975px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the normalized distance from the well, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ey\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8.94167px 7.91667px; transform-origin: 8.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = solveODEB(x,coeff,Xbc,bc)\r\n%  y = values of the solution\r\n%  x = values of the independent variable where the solution is requested\r\n%  coeff = [a p], the two parameters in the ODE\r\n%  Xbc = values of x where the boundary conditions are specifified\r\n%  bc = [y0 yp0 y1] = boundary values (see description)\r\n\r\n  y = f(x,coeff,Xbc,bc);\r\nend","test_suite":"%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [1 NaN 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [1 0.505461057363874 0.265959532078956 0.140787844664873 0.071276733472728 0.029333752731051 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [0 NaN 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000035290165611 0.000065476121971 0.000315436935125 0.000552779648598 0.000757412623201 0.003086031722519 0.012031119481478 0.040129255417251 0.089700339919646];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [4 NaN -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [2.974028994378906 1.041176175714286 0.308462205553512 -0.076175488441917 -0.426089583908447 -0.952692417292867];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [2 4];\r\nXbc = [1 Inf];\r\nbc = [3 NaN 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [3 0.005688558853080 0.000051725255081 0.000000653418086 0.000000009396692];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n%-------------------------------\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 4];\r\nbc = [NaN 1 0];\r\nx = linspace(1,4,7);\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-0.696760997897786 -0.352185550727323 -0.185310228971762 -0.098095479140575 -0.049662847941353 -0.020438614824974 0];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n \r\n%%\r\ncoeff = [3 1];\r\nXbc = [0.5 4.5];\r\nbc = [NaN 0 2];\r\nx = [cos(pi/4) cos(pi/6) sqrt(2) (1+sqrt(5))/2 sqrt(3) sqrt(5) exp(1) pi 4+psi(1)];\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [0.000053997354167 0.000075728700374 0.000316920386259 0.000553525214653 0.000757922298088 0.003086129240342 0.012031140116004 0.040129260776539 0.089700342119009];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 1/2];\r\nXbc = [1 5];\r\nbc = [NaN 4 -1];\r\nx = 1.2:0.75:4.95;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.047695617799804 -0.816404083826666 -0.431398160565887 -0.374418157507686 -0.532801281305956 -0.958228725044217];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n\r\n%%\r\ncoeff = [1 0];\r\nXbc = [1 Inf];\r\nbc = [NaN 3 0];\r\nx = 1:2:9;\r\ny = solveODEB(x,coeff,Xbc,bc);\r\ny_correct = [-2.098451806781317 -0.173147136186896 -0.018397012773047 -0.002117248575316 -0.000253600440751];\r\nassert(all(abs(y-y_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-16T01:10:22.000Z","updated_at":"2025-05-09T06:39:10.000Z","published_at":"2021-05-16T01:19:31.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve the following ordinary differential equation:  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\u0026quot; + (1/x) y' - (a^2 + p^2/x^2) y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{d^2y}{dx^2} +\\\\frac{1}{x} \\\\frac{dy}{dx}-\\\\left(a^2+\\\\frac{p^2}{x^2}\\\\right)y = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x1) = y1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_1) = y_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and either \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y(x0) = y0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey(x_0) = y_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y'(x0) = y'0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\\\\prime(x_0) = y\\\\prime_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Along with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, one of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eyp0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will be assigned numerical values, and the other will be \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNaN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The function should return the values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at the specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne of the applications for this equation is in groundwater. For \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"p = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ep = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the equation arises in the flow of water to a well pumping in a leaky confined aquifer; the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the normalized distance from the well, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is related to the piezometric head, a combination of the elevation and pressure of the water. Specifying the derivative at a point amounts to specifying the flow to the well.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"differential 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