| course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
53 |
|
Paper 4, Section II, A |
2009 |
Suppose that $y_{1}(x)$ and $y_{2}(x)$ are linearly independent solutions of
$$\frac{d^{2} y}{d x^{2}}+b(x) \frac{d y}{d x}+c(x) y=0$$
with $y_{1}(0)=0$ and $y_{2}(1)=0$. Show that the Green's function $G(x, \xi)$ for the interval $0 \leqslant x, \xi \leqslant 1$ and with $G(0, \xi)=G(1, \xi)=0$ can be written in the form
$$G(x, \xi)= \begin{cases}y_{1}(x) y_{2}(\xi) / W(\xi) ; & 0<x<\xi \ y_{2}(x) y_{1}(\xi) / W(\xi) ; & \xi<x<1\end{cases}$$
where $W(x)=W\left[y_{1}(x), y_{2}(x)\right]$ is the Wronskian of $y_{1}(x)$ and $y_{2}(x)$.
Use this result to find the Green's function $G(x, \xi)$ that satisfies
$$\frac{d^{2} G}{d x^{2}}+3 \frac{d G}{d x}+2 G=\delta(x-\xi)$$
in the interval $0 \leqslant x, \xi \leqslant 1$ and with $G(0, \xi)=G(1, \xi)=0$. Hence obtain an integral expression for the solution of
$$\frac{d^{2} y}{d x^{2}}+3 \frac{d y}{d x}+2 y= \begin{cases}0 ; & 0<x<x_{0} \ 2 ; & x_{0}<x<1\end{cases}$$
for the case $x<x_{0}$.