Sturm-Liouville eigenfunctions and the equations that define the eigenvalues

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)