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Orthogonality functions

  1. Mar 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that the eigenfunctions of (1) are orthogonal.

    [tex]
    m y_{tt} + 2Uy_{tx} + U^2y_{xx} +EIy_{xxxx} =0[/tex] (1)

    for

    [tex]
    0<x<L, t>0
    [/tex]

    with

    [tex]
    y(0,t) = 0
    [/tex][tex]
    y_x(x,t) = 0
    [/tex][tex]
    y_{xx}(L,t) = 0
    [/tex][tex]
    y_{xxx}(L,t) = 0
    [/tex]
    [tex]
    y(x,0) = f(x)
    [/tex][tex]
    y_t (x,0) = g(x)
    [/tex]


    3. The attempt at a solution
    I substituted:

    [tex]
    y= e^{-\lambda_n t} X_n(x)
    [/tex] and[tex]
    y= e^{-\lambda_m t} X_m(x)
    [/tex] and subtracted the results.

    The problem lies in the mixed derative term, i can not seem to let it vanish with the help of the BC's:

    [tex]
    \lambda_n X_m X'_n - \lambda_m X_n X'_m
    [/tex]

    How am I suppose to let that term vanish? I've tried IBP but iw will not vanish.
     
  2. jcsd
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