Find the eigenvalues and eigenfunctions for $$y'' + \lambda y = 0, y(0) = 0, y'(\pi/2) = 0$$
According to my book we must check 3 cases: $\lambda < 0$, $\lambda = 0$, $\lambda > 0$.
I started with $\lambda > 0$, found the general solution, applied the boundary conditions, and found $$\lambda _n = (2n-1)^2 with~ y_n=sin((2n-1)x)$$
I moved on to $\lambda < 0$ and got $$y = c_1e^{kx}+c_2e^{-kx}$$ and didn't know how to proceed.
Best Answer
Let $\lambda = -\mu^2$; then
$$y(x) = A e^{\mu x} + B e^{-\mu x}$$
$y(0)=0 \implies A+B=0$. Also,
$$y'\left( \frac{\pi}{2}\right) = 0 \implies 2 A \sinh{\left(\mu \frac{\pi}{2}\right)} =0 $$
The only way this happens is when $A=0$; that is, there is no nontrivial solution to this BVP for $\lambda < 0$.