Dirichlet half-line response to uniform interval forcing

For $\mu >0$, the decaying solution of $-y^{\prime \prime }+y=1_{[0,\mu ]}$ on the positive half-line with $y(0)=0$ follows by integrating the Dirichlet half-line Green function for minus d2 plus 1:Both $y$ and $y^{\prime }$ are continuous at $\mu$, whereas $y^{\prime \prime }$ has the jump required by the bounded step in the forcing. Its common value there is $(1-e^{-\mu }{)}^2/2$.