(a) Determine the Green's function $G(x ; \xi)$ satisfying

$$G^{\prime \prime}-4 G^{\prime}+4 G=\delta(x-\xi),$$

with $G(0 ; \xi)=G(1 ; \xi)=0$. Here ' denotes differentiation with respect to $x$.

(b) Using the Green's function, solve

$$y^{\prime \prime}-4 y^{\prime}+4 y=e^{2 x}$$

with $y(0)=y(1)=0$.