Doubt regarding Eigen values and Eigen functions of a boundary value problem.

Question

Consider the differential equation ${{d^2y}\over{dx^2}}+\lambda y=0,\lambda>0$ where $y(-l)=y(l)=0,l>0$. Now from the two boundary conditions I am getting $C_1\cos(\sqrt{\lambda}x)-C_2\sin(\sqrt{\lambda}x)=0$ and $C_1\cos(\sqrt{\lambda}x)+C_2\sin(\sqrt{\lambda}x)=0$,from here we obtain by addition that $C_1=0$ or $\cos(\sqrt{\lambda}x)=0$,but $C_1=0$ implies $\sin(\sqrt{\lambda}x)=0$ considering non-trivial solution. So we have finally $\sin(\sqrt{\lambda}x)=0$ or $\cos(\sqrt{\lambda}x)=0$,but the answer to the problem is given as $\lambda_n=n^2\pi^2$,$n\in \mathbb N$.So,what has gone wrong here?Am I missing something?

Answer 1

The given answer is obviously wrong. From the conditions you get that $\sqrtλl=n\frac\pi2$ for some $n\ge 0$ so that $$λ_n=\frac{n^2\pi^2}{4l^2}.$$

You could unify both solution branches as $y(x)=C\sin(\sqrtλ(x+l))$ so that $y(-l)=0$ automatically and $0=y(l)=\sin(2\sqrtλl)$ again implies $2\sqrtλl=n\pi$.