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Math Help - Eigenvalues and eigenfunctions

  1. #1
    Senior Member chella182's Avatar
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    Eigenvalues and eigenfunctions

    Find the eigenvalues and eigenfunctions for

    y''+\lambda y=0, 0\leq x\leq1

    with the homogenous boundary conditions y(0)=y'(1)=0.



    So I subbed in y=e^{\alpha x} et cetera, and somewhere down the line ended up needing to solve

    \sin{\omega}+\omega\cos{\omega}=0

    and now I'm stuck and not even sure if this can be done. Obviously \omega=0 would work, but wouldn't that just give a trivial solution?
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by chella182 View Post
    Find the eigenvalues and eigenfunctions for

    y''+\lambda y=0, 0\leq x\leq1

    with the homogenous boundary conditions y(0)=y'(1)=0.



    So I subbed in y=e^{\alpha x} et cetera, and somewhere down the line ended up needing to solve

    \sin{\omega}+\omega\cos{\omega}=0

    and now I'm stuck and not even sure if this can be done. Obviously \omega=0 would work, but wouldn't that just give a trivial solution?
    I assume that you have put \omega = \sqrt{-\lambda}, so that the general solution of the equation is y = A\cos\omega x + B\sin\omega x. The boundary condition y(0)=0 tells you that A=0, so that y = B\sin\omega x. Then you differentiate to get the other boundary condition in the form \omega\cos\omega = 0. You can disregard the solution \omega=0, because that leads to the trivial solution y=0. So the eigenvalues are given by \cos\omega=0, which has an infinite family of solutions.
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  3. #3
    Senior Member chella182's Avatar
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    Ignore this, I've figured out why you don't have e^{x} in your solution. Damn shoddy notes :@
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