Consider the differential operator

$$\mathcal{L} y=\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}$$

acting on real functions $y(x)$ with $0 \leqslant x \leqslant 1$.

(i) Recast the eigenvalue equation $\mathcal{L} y=-\lambda y$ in Sturm-Liouville form $\tilde{\mathcal{L}} y=-\lambda w y$, identifying $\tilde{\mathcal{L}}$ and $w$.

(ii) If boundary conditions $y(0)=y(1)=0$ are imposed, show that the eigenvalues form an infinite discrete set $\lambda_{1}<\lambda_{2}<\ldots$ and find the corresponding eigenfunctions $y_{n}(x)$ for $n=1,2, \ldots$. If $f(x)=x-x^{2}$ on $0 \leqslant x \leqslant 1$ is expanded in terms of your eigenfunctions i.e. $f(x)=\sum_{n=1}^{\infty} A_{n} y_{n}(x)$, give an expression for $A_{n}$. The expression can be given in terms of integrals that you need not evaluate.