# Math Help - Sturm Liouville System

1. ## Sturm Liouville System

$x d^2/dx^2-dy/dx=-lambda.x^3y$

$y(0)=0, y(a)=0$

Find the eigenvalues and corresponding eigenfunctions of the given BVP.

So, through substitution, I found that the given equation transforms to the simple harmonic equation:

$y''+(lamdba/4)y=0$

where

y=At+B when lambda=0
$y=Ce^((sqrt(-lambda/4))t)+De^(-(sqrt(-lambda/4))t)$ when lambda<0 *sorry, can't get the latex to work for me: it's meant to be to the power of e
$y=Ecos(sqrt(-lambda/4))t)+Fsin(sqrt(-lambda/4))t)$ lambda>0

I then put the equation in the form of a sturm liouville system by multiplying by factor $k=1/(x^2)$ such that I got: $-(1/x.y')'=lambda.x.y$

However, I was then unsure of how to go about getting the required eigenvalues and eigenfunctions due to the y(a)=0 boundary condition. Thanks in advance.

2. Or even for the general case sturm liouville system, how do you go about getting the eigenvalues and eigenfunctions? I really do not understand the lecture notes on this topic :P