= Solution
Away from $x=\xi$, the <Green function> solves $G''-G=0$. The solution satisfying the left <boundary condition> is proportional to $\sinh x$, while the solution decaying at infinity is proportional to $e^{-x}$. Continuity at $\xi$ and the unit <derivative> jump
$$
G_x(\xi+;\xi)-G_x(\xi-;\xi)=1
$$
give
$$
\boxed{
G(x;\xi)=
\begin{cases}
-e^{-\xi}\sinh x,&0<x<\xi,\\
-e^{-x}\sinh\xi,&x>\xi,
\end{cases}
}
$$
or $G=-\sinh(\min\{x,\xi\})e^{-\max\{x,\xi\}}$, the <dirichlet half-line Green function for d2 minus 1>.
The required solution is
$$
y(x)=\int_0^\infty G(x;\xi)e^{-2\xi}\,d\xi.
$$
Evaluation of the two elementary <integrals>, split at $\xi=x$, gives
$$
\boxed{y(x)=\frac13(e^{-2x}-e^{-x})}.
$$
Indeed, direct <differentiation> gives $y''-y=e^{-2x}$, and both <boundary conditions> are immediate.
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