Show that a smooth function $y(x)$ that satisfies $y(0)=y^{\prime}(1)=0$ can be written as a Fourier series of the form

$$y(x)=\sum_{n=0}^{\infty} a_{n} \sin \lambda_{n} x, \quad 0 \leqslant x \leqslant 1$$

where the $\lambda_{n}$ should be specified. Write down an integral expression for $a_{n}$.

Hence solve the following differential equation

$$y^{\prime \prime}-\alpha^{2} y=x \cos \pi x$$

with boundary conditions $y(0)=y^{\prime}(1)=0$, in the form of an infinite series.

$$\begin{aligned} & \text { (i) } f(x)=e^{-a|x|} \text {, } \\ & \text { and }(i i) \quad g(x)= \begin{cases}1, & |x| \leqslant b \\ 0, & |x|>b .\end{cases} \end{aligned}$$