Solution
= Solution
Write $G(x;\xi)=e^{2x}H(x;\xi)$. Since $(D-2)^2(e^{2x}H)=e^{2x}H''$, the <Green function> must have $H(0;\xi)=H(1;\xi)=0$, be continuous at $x=\xi$, and satisfy the <jump condition for an impulse> $H_x(\xi^+;\xi)-H_x(\xi^-;\xi)=e^{-2\xi}$. Solving the two linear pieces gives
$$
\boxed{
G(x;\xi)=e^{2(x-\xi)}
\begin{cases}
x(\xi-1),&0\leq x\leq\xi,\\
\xi(x-1),&\xi\leq x\leq1.
\end{cases}}
$$