How to find eigenvalues and eigenfunctions of this boundary value problem? $$ y'' + \lambda y = 0 \\ y'(0)=0, y(\pi/2)=0 $$ I want to find only positive eigenvalues. I proceed like this: $$ y=C_1 \cos(\sqrt{\lambda} x) + C_2 \sin(\sqrt{\lambda} x)\\ y(\pi/2)=0 \Rightarrow C_2=0\\ \therefore y=C_1 \cos(\sqrt{\lambda }x \\ y'=-C_1 \sqrt{\lambda } \sin(\sqrt{\lambda} x) \\ y'(0)=0 \Rightarrow -C_1 \sqrt{\lambda } \sin(\sqrt{\lambda} x) $$ So, $C_1=0$ or $\lambda=0$. In these cases, we only get trivial solution.
So, does any value of $ \lambda $ is eigenvalue? Or have I made errors?