course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
37 |
|
4.II.16C |
2004 |
Obtain the Green function $G(x, \xi)$ satisfying
$$G^{\prime \prime}+\frac{2}{x} G^{\prime}+k^{2} G=\delta(x-\xi),$$
where $k$ is real, subject to the boundary conditions
$$\begin{array}{rll}
G \text { is finite } & \text { at } & x=0, \\
G=0 & \text { at } & x=1 .
\end{array}$$
[Hint: You may find the substitution $G=H / x$ helpful.]
Use the Green function to determine that the solution of the differential equation
$$y^{\prime \prime}+\frac{2}{x} y^{\prime}+k^{2} y=1,$$
subject to the boundary conditions
$$\begin{array}{rll}
y \text { is finite } & \text { at } & x=0, \\
y=0 & \text { at } & x=1,
\end{array}$$
is
$$y=\frac{1}{k^{2}}\left[1-\frac{\sin k x}{x \sin k}\right]$$