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Sturm-Liouville problem

  1. Jun 13, 2008 #1
    1. The problem statement, all variables and given/known data
    Find the eigenvalues and the eigenfunctions of the Sturm-Liouville problem

    [tex] \frac{d^{2}u}{dx^{2}}=\lambda u [/tex]
    [tex] 0<x<L[/tex]
    [tex]\frac{du}{dx}(0) = 0[/tex]
    [tex]u(L) = 0[/tex]

    3. The attempt at a solution
    characteristic polynomial:
    [tex] p^{2}=+-\lambda[/tex]
    [tex]u = Ae^{\sqrt{\lambda}x}+Be^{-\sqrt{\lambda}x}[/tex]
    [tex]u = Ccosh(\sqrt{\lambda}x)+Dsinh(-\sqrt{\lambda}x)[/tex]

    Now, i try to solve the boundaries:
    [tex]
    \frac{du}{dx}(0)=-D\sqrt{\lambda}cosh(-\sqrt{\lambda}x)=0
    [/tex] ... im confused now because cosh doesn't have a root unless its translated. Can anyone help me out with this please?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 13, 2008 #2

    tiny-tim

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    Homework Helper

    Hi EngageEngage!

    Isn't it just D = 0? :smile:
     
  4. Jun 13, 2008 #3
    O yeah, thanks. not sure how i managed to screw that up. then I get
    [tex] u = Ccosh(\sqrt{\lambda x})[/tex]
    [tex] u(L) = 0 = Ccosh(\sqrt{\lambda L})[/tex]
    But cosh has no root here so, i get u = 0, by setting C = 0.
    I just realized it: i probably have to do this with lambda>0, <0 and = 0, is that righT?
     
  5. Jun 13, 2008 #4
    so when: [tex] \lambda = 0[/tex]
    [tex]
    u = C(1)+D(0);;;
    u = C
    [/tex]
    and then i get u = 0 again.

    now when [tex] \lambda <0[/tex]
    [tex] u = Ae^{i\sqrt{|\lambda|}x}+Be^{-i\sqrt{|\lambda|}x}[/tex]
    [tex] u = Ecos(\sqrt{\lambda}x)+Fsin(\sqrt{\lambda}x)[/tex]
    [tex] u'(0)=0 when F = 0[/tex]
    [tex] u = Ecos(\sqrt{\lambda}x)[/tex]
    [tex] u(L)=0=Ecos(\sqrt{\lambda}L)[/tex]
    [tex] \lambda = (\frac{(n-\frac{1}{2})\pi}{L})^{2}, n = 1,2,3...[/tex]
    [tex] u = cos(\frac{(n-\frac{1}{2})\pi x}{L})[/tex]
     
    Last edited: Jun 13, 2008
  6. Jun 13, 2008 #5
    is that right?
     
  7. Jun 13, 2008 #6
    messed that last part up, but i got it now. Thanks!
     
  8. Jun 14, 2008 #7

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi EngageEngage! :smile:

    Yes, that looks fine!

    (though personally I don't like minuses, so I'd write it:

    [tex] u = cos\left(\frac{(n+\frac{1}{2})\pi x}{L}\right)[/tex] )

    (note the \left( and \right) in the LaTeX! :smile: )
     
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