Solution
= Solution
The <Green-function representation> gives
$$
y(x)=\int_0^1G(x;\xi)e^{2\xi}\,d\xi
=e^{2x}\left[(x-1)\int_0^x\xi\,d\xi+x\int_x^1(\xi-1)\,d\xi\right].
$$
The bracket simplifies to $x(x-1)/2$, so
$$
\boxed{y(x)=\frac12x(x-1)e^{2x}.}
$$
It vanishes at both endpoints, and applying $(D-2)^2$ returns $e^{2x}$.