Consider the real Sturm-Liouville problem

$$\mathcal{L} y(x)=-\left(p(x) y^{\prime}\right)^{\prime}+q(x) y=\lambda r(x) y,$$

with the boundary conditions $y(a)=y(b)=0$, where $p, q$ and $r$ are continuous and positive on $[a, b]$. Show that, with suitable choices of inner product and normalisation, the eigenfunctions $y_{n}(x), \quad n=1,2,3 \ldots$, form an orthonormal set.

Hence show that the corresponding Green's function $G(x, \xi)$ satisfying

$$(\mathcal{L}-\mu r(x)) G(x, \xi)=\delta(x-\xi),$$

where $\mu$ is not an eigenvalue, is

$$G(x, \xi)=\sum_{n=1}^{\infty} \frac{y_{n}(x) y_{n}(\xi)}{\lambda_{n}-\mu}$$

where $\lambda_{n}$ is the eigenvalue corresponding to $y_{n}$.

Find the Green's function in the case where

$$\mathcal{L} y \equiv y^{\prime \prime},$$

with boundary conditions $y(0)=y(\pi)=0$, and deduce, by suitable choice of $\mu$, that

$$\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .$$