(a) Find the Fourier sine series of the function

$$f(x)=x$$

for $0 \leqslant x \leqslant 1$.

(b) The differential operator $L$ acting on $y$ is given by

$$L[y]=y^{\prime \prime}+y^{\prime}$$

Show that the eigenvalues $\lambda$ in the eigenvalue problem

$$L[y]=\lambda y, \quad y(0)=y(1)=0$$

are given by $\lambda=-n^{2} \pi^{2}-\frac{1}{4}, \quad n=1,2, \ldots$, and find the corresponding eigenfunctions $y_{n}(x)$.

By expressing the equation $L[y]=\lambda y$ in Sturm-Liouville form or otherwise, write down the orthogonality relation for the $y_{n}$. Assuming the completeness of the eigenfunctions and using the result of part (a), find, in the form of a series, a function $y(x)$ which satisfies

$$L[y]=x e^{-x / 2}$$

and $y(0)=y(1)=0$.