| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
39 |
|
2.II.15G |
2006 |
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.