Verify that $y=e^{-x}$ is a solution of the differential equation

$$(x+2) y^{\prime \prime}+(x+1) y^{\prime}-y=0,$$

and find a second solution of the form $a x+b$.

Let $L$ be the operator

$$L[y]=y^{\prime \prime}+\frac{(x+1)}{(x+2)} y^{\prime}-\frac{1}{(x+2)} y$$

on functions $y(x)$ satisfying

$$y^{\prime}(0)=y(0) \quad \text { and } \quad \lim _{x \rightarrow \infty} y(x)=0 .$$

The Green's function $G(x, \xi)$ for $L$ satisfies

$$L[G]=\delta(x-\xi)$$

with $\xi>0$. Show that

$$G(x, \xi)=-\frac{(\xi+1)}{(\xi+2)} e^{\xi-x}$$

for $x>\xi$, and find $G(x, \xi)$ for $x<\xi$.

Hence or otherwise find the solution of

$$L[y]=-(x+2) e^{-x},$$

for $x \geqslant 0$, with $y(x)$ satisfying the boundary conditions above.