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Math Help - [SOLVED] eigenvalue problem

  1. #1
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    [SOLVED] eigenvalue problem

    I want to find the eigenvalues and the eigenfunctions of the operator

     ({d^{2}}/{dx^{2}})+k

    so we have the problem

     ({d^{2}y_{n}}/{dx^{2}})+ky_{n}=L_{n} y_{n}

    Now I put in the functie y=A*exp(n*x), is it correct to do this?, and get the equation

     n^{2}+k-L_{n} =0

    so Ln=n^2+k

    but acording to the solutions it has to be Ln=n^2-k. Where do I make the mistake?
    and how do I exactly determine the eigenfunctions

    thanks
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  2. #2
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    Quote Originally Posted by charlie123 View Post
    I want to find the eigenvalues and the eigenfunctions of the operator

     ({d^{2}}/{dx^{2}})+k

    so we have the problem

     ({d^{2}y_{n}}/{dx^{2}})+ky_{n}=L_{n} y_{n}

    Now I put in the functie y=A*exp(n*x), is it correct to do this?, and get the equation

     n^{2}+k-L_{n} =0

    so Ln=n^2+k

    but acording to the solutions it has to be Ln=n^2-k. Where do I make the mistake?
    and how do I exactly determine the eigenfunctions
    This problem as stated is incomplete. The differential operator is not completely defined unless some boundary conditions are specified. If the boundary condition says that y→∞ as x→∞ then you are correct to take functions of the form A*exp(n*x) as eigenfunctions. More often, the boundary conditions will say for example y=0 when x = 0 and when x = π. In that case, the eigenfunctions will be of the form A*sin(nx) and the eigenvalues will be kľn^2 (still not equal to the given solution n^2ľk, but perhaps you are using a different convention about signs).
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  3. #3
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    the boundary conditions are y(0)=0 and y(pi)=0

    but how do I know what the the eigenfunction is? Here I can see it of course, but if I didn't know it for instance how do I start solving this problem?
    The determining of the eigenvalue is not difficult after that.

    thanks for the help
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  4. #4
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    Quote Originally Posted by charlie123 View Post
    the boundary conditions are y(0)=0 and y(pi)=0

    but how do I know what the the eigenfunction is? Here I can see it of course, but if I didn't know it for instance how do I start solving this problem?
    The determining of the eigenvalue is not difficult after that.
    The differential equation is d^2y/dx^2 + (k-\lambda)y = 0, where \lambda is an eigenvalue. This is a standard SHM equation, and I guess you're supposed to know that the general solution is y = A\sin\omega x + B\cos\omega x, where \omega^2 = k-\lambda. The boundary condition y(0)=0 tells you that B=0, and the boundary condition y(π)=0 tells you that \omega must be an integer, say n. Therefore k-\lambda = n^2.

    So you see it's not a question of finding the eigenfunction first and then the eigenvalue. They both emerge together as a result of solving the differential equation together with its boundary conditions.
    Last edited by Opalg; February 9th 2009 at 12:52 AM. Reason: typo
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