Sturm-Liouville eigenfunctions and the equations that define the eigenvalues

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Hey so I'm trying to solve this Sturm-Liouville problem. Here is something I've written so far but I'm not sure how I can apply y(0)=0, and y(1)+y'(1)=0 to y and its derivatives for each case. Please if you have any suggestions on how I can tackle this problem I'd appreciate it.
% Problem_2 the Sturm-Liuoville Problem
% y" + lambda * y = 0, y'(0) = 0, y(1) + y'(1) = 0
% consider the cases lambda = 0, lambda = -alpha^2 > 0, lambda = alpha^2 > 0
% Case (1): lambda = 0 the eq. becomes y"=0
syms x C1 C2
y=C1*x+C2;
diff(y)
% Case (2): lambda = -alpha^2 > 0 the eq. becomes y" - alpha^2 * y = 0
syms x C1 C2 alpha
y1=C1*exp(alpha*x)+C2*exp(-alpha*x);
diff(y1)
% Case (3): lambda = alpha^2 > 0 the eq. becomes y" + alpha^2 * y = 0
syms x C1 C2 alpha
y2=C1*cos(alpha*x)+C2*sin(alpha*x);
diff(y2)

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