Consider the linear differential operator $\mathcal{L}$ defined by

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

on the interval $0 \leqslant x<\infty$. Given the boundary conditions $y(0)=0$ and $\lim _{x \rightarrow \infty} y(x)=0$, find the Green's function $G(x, \xi)$ for $\mathcal{L}$ with these boundary conditions. Hence, or otherwise, obtain the solution of

$$\mathcal{L} y= \begin{cases}1, & 0 \leqslant x \leqslant \mu \\ 0, & \mu<x<\infty\end{cases}$$

subject to the above boundary conditions, where $\mu$ is a positive constant. Show that your piecewise solution is continuous at $x=\mu$ and has the value

$$y(\mu)=\frac{1}{2}\left(1+e^{-2 \mu}-2 e^{-\mu}\right) .$$