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Vector ODE using forward Euler method

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Liam Quiros Edmunds
Liam Quiros Edmunds on 10 Jan 2021
Answered: James Tursa on 10 Jan 2021
Having trouble defining the following vector ode in matlab.
I have written the following for the right hand side function for suitable use in ode45. The system models a projectile with position and velocity vectors (x(t), y(t)) and (u(t), v(t)) at time t with parameters g and mu for gravity and air resistance.
function dydt = rhs_1(t, y, g, mu)
dxdt1 = u;
dydt2 = v;
dudt3 = -mu.*y(1).*sqrt(y(1).^2+y(2).^2);
dvdt4 = -g-mu.*y(2).*sqrt(y(1).^2+y(2).^2);
dydt = [dxdt1, dydt2, dudt3, dvdt4];
end
And when I run the following;
[t,y,g,mu]=ode45(@rhs_1,[0 100],[0.1; 0], 9.8, 0.5);
plot(t,y,g,mu(:,1),'r.-',t,y(:,2),'g.-','LineWidth',2,'MarkerSize',10);
I get the following error messages;
Unrecognized function or variable 'u'.
Error in rhs_1 (line 2)
dxdt1 = u;
Error in odearguments (line 90)
f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 115)
odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);
Error in runode (line 1)
[t,y,g,mu]=ode45(@rhs_1,[0 100],[0.1; 0], 9.8, 0.5);
Apologies if this is trivial, I am new to Matlab. Any help is greatly appreciated. Thank you!

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Answers (2)

Jan
Jan on 10 Jan 2021
Edited: Jan on 10 Jan 2021
You have replaced the "x" of the formula by "y(1)", the "y" by "y(2)". Now you have to replace "u" by "y(3)" also and "v" by "y(4)":
function dydt = rhs_1(t, y, g, mu)
dxdt1 = y(3); % instead of u
dydt2 = y(4); % instead of v
dudt3 = -mu .* y(3) .* sqrt(y(3).^2 + y(4).^2); % y(1) -> y(3), y(2) -> y(4)
dvdt4 = -g - mu .* y(4) .* sqrt(y(3).^2 + y(4).^2);
dydt = [dxdt1, dydt2, dudt3, dvdt4];
end

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James Tursa
James Tursa on 10 Jan 2021
Also your main code should look something like this:
g = 9.8;
mu = 0.5;
[t,y]=ode45(@(t,y)rhs_1(t,y,g,mu),[0 100],[0.1; 0]);
Then, depending on how you want your plotting, something like this:
plot(y(:,1),y(:,2));
or
plot(t,y(:,1),t,y(:,2));

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