course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
39 |
|
2.II.15D |
2007 |
Let $y_{0}(x)$ be a non-zero solution of the Sturm-Liouville equation
$$L\left(y_{0} ; \lambda_{0}\right) \equiv \frac{d}{d x}\left(p(x) \frac{d y_{0}}{d x}\right)+\left(q(x)+\lambda_{0} w(x)\right) y_{0}=0$$
with boundary conditions $y_{0}(0)=y_{0}(1)=0$. Show that, if $y(x)$ and $f(x)$ are related by
$$L\left(y ; \lambda_{0}\right)=f$$
with $y(x)$ satisfying the same boundary conditions as $y_{0}(x)$, then
$$\int_{0}^{1} y_{0} f d x=0$$
Suppose that $y_{0}$ is normalised so that
$$\int_{0}^{1} w y_{0}^{2} d x=1$$
and consider the problem
$$L(y ; \lambda)=y^{3} ; \quad y(0)=y(1)=0$$
By choosing $f$ appropriately in $(*)$ deduce that, if
$$\lambda-\lambda_{0}=\epsilon^{2} \mu[\mu=O(1), \epsilon \ll 1], \quad \text { and } \quad y(x)=\epsilon y_{0}(x)+\epsilon^{2} y_{1}(x)$$
then
$$\mu=\int_{0}^{1} y_{0}^{4} d x+O(\epsilon)$$