= Dirichlet half-line response to uniform interval forcing
For $\mu>0$, the decaying solution of $-y''+y=\mathbf1_{[0,\mu]}$ on the positive half-line with $y(0)=0$ follows by integrating the <Dirichlet half-line Green function for minus d2 plus 1>:
$$
y(x)=\begin{cases}1-e^{-x}-e^{-\mu}\sinh x,&0\leq x\leq\mu,\\(\cosh\mu-1)e^{-x},&x\geq\mu.\end{cases}
$$
Both $y$ and $y'$ are continuous at $\mu$, whereas $y''$ has the jump required by the bounded step in the forcing. Its common value there is $(1-e^{-\mu})^2/2$.
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