(i) Let $f(x)=x, 0<x \leqslant \pi$. Obtain the Fourier sine series and sketch the odd and even periodic extensions of $f(x)$ over the interval $-2 \pi \leqslant x \leqslant 2 \pi$. Deduce that

$$\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$

(ii) Consider the eigenvalue problem

$$\mathcal{L} y=-\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}=\lambda y, \quad \lambda \in \mathbb{R}$$

with boundary conditions $y(0)=y(\pi)=0$. Find the eigenvalues and corresponding eigenfunctions. Recast $\mathcal{L}$ in Sturm-Liouville form and give the orthogonality condition for the eigenfunctions. Using the Fourier sine series obtained in part (i), or otherwise, and assuming completeness of the eigenfunctions, find a series for $y$ that satisfies

$$\mathcal{L} y=x e^{-x}$$

for the given boundary conditions.