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Thread: 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

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

    with

    $\displaystyle x(0) = x(1) = 0$.

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

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

    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, $\displaystyle r^2+ 2r+ \lambda= 0$.

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

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

    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 $\displaystyle \lambda > 1 $

    and that are given by:

    $\displaystyle x_{n}(t)=Ce^{-t}\sin((\sqrt{\lambda_{n}-1})t), \mbox{ where } \lambda_{n} = n^2\pi^2 + 1 \mbox{ and } C \in \mathbb{R}$
    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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