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Math Help - Regular S-L problem

  1. #1
    Member Ruun's Avatar
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    Regular S-L problem

    Hi, I'm trying to find the eigenfunctions and eigenvalues of

    x''(t)+2x'(t)+\lambda x(t) = 0

    with

    x(0) = x(1) = 0.

    First, should I write in self adjoint form? I found it is

    (e^{2t}x'(t))' + \lambda x(t) = 0

    But I don't know how to continue, all the previous examples ended in a harmonic oscillator type equation. Just a hint will be enough

    Thanks.
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  2. #2
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    I would just solve the characteristic equation, r^2+ 2r+ \lambda= 0.

    r= \frac{-2\pm\sqrt{4- 4\lambda}}{2}= -1\pm\sqrt{1- \lambda}

    Now look at what happens for \lambda< 1, \lambda= 1, \lambda> 1.

    Which of those give functions that will satisfy the boundary conditions non-trivially?
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  3. #3
    Member Ruun's Avatar
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    Hi, thanks for your post!

    Well, I found that we have non-trivial solutions only if \lambda > 1

    and that are given by:

    x_{n}(t)=Ce^{-t}\sin((\sqrt{\lambda_{n}-1})t), \mbox{ where } \lambda_{n} = n^2\pi^2 + 1 \mbox{ and } C \in \mathbb{R}
    Last edited by Ruun; May 24th 2011 at 06:27 AM. Reason: typo
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  4. #4
    MHF Contributor

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    Yes, that is exactly right!
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