Problem source

The function $y(x)$ is  defined for $x\ge0$ and satisfies the conditions
 \[
y=0
\mbox{ \ \  and \ \ }
\frac{\d y}{\d x}=1
\mbox{ \  \  at $x=0$}.
\]
When $x$ is in the range $2(n-1)\pi< x <2n\pi$, where $n$ is a positive 
integer, $y(t)$ satisfies the differential
equation 
$$
{\d^2y \over \d x^2} + n^2 y=0. 
$$
Both $y$ and $\displaystyle \frac{\d y}{\d x} $ are continuous at $x=2n\pi$ for 
$n=0,\; 1,\;2,\; \ldots\;$. 
\begin{questionparts}
\item Find $y(x)$ for $0\le x \le 2\pi$. 
\item Show that $y(x) = \frac12 \sin 2x $ for 
$2\pi\le x\le 4\pi$, and find $y(x)$ for all $x\ge0$.
\item Show that 
$$
\int_0^\infty y^2 \,\d x = \pi \sum_{n=1}^\infty {1\over n^2} \,.
$$
\end{questionparts}