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

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

Find the eigenvalues and corresponding eigenfunctions of the given BVP.

Answer:

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

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

where

y=At+B when lambda=0

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

$\displaystyle y=Ecos(sqrt(-lambda/4))t)+Fsin(sqrt(-lambda/4))t)$ lambda>0

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

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.