Calculate the Green's function $G(x ; \xi)$ given by the solution to

$$\frac{d^{2} G}{d x^{2}}=\delta(x-\xi) ; \quad G(0 ; \xi)=\frac{d G}{d x}(1 ; \xi)=0$$

where $\xi \in(0,1), x \in(0,1)$ and $\delta(x)$ is the Dirac $\delta$-function. Use this Green's function to calculate an explicit solution $y(x)$ to the boundary value problem

$$\frac{d^{2} y}{d x^{2}}=x e^{-x}$$

where $x \in(0,1)$, and $y(0)=y^{\prime}(1)=0$.