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Math Help - Sturm Liouville System

  1. #1
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    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.

    Answer:

    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.
    Last edited by TheFirstOrder; May 30th 2010 at 02:01 AM. Reason: Typo
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  2. #2
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    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
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