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Homework Help: Finding Eigen Values/functions

  1. Sep 3, 2010 #1
    1. The problem statement, all variables and given/known data
    I have to find the eigenvalues&function of the eqn:
    [tex]y''+\lambda y =0[/tex]
    With the boundary conditions:
    [tex]y(0)+y'(0) = 0[/tex] and [tex]y(\pi) =0[/tex]

    2. Relevant equations



    3. The attempt at a solution
    I get the general equations, okay, but am having trouble due to the boundary conditions.
    Assuming [tex]\lambda>0[/tex], then get the general solution:
    [tex]y(x)=A\sin(x\sqrt{\lambda})+B\cos(x\sqrt{\lambda})[/tex]

    The best i can do now is that:
    [tex]y(\pi)=A\sin(\sqrt{\lambda}\pi)+B\cos(\sqrt{\lambda}\pi)=0[/tex]
    Which can only be valid if B = 0 and [tex]\sqrt{\lambda} = n[/tex] where n is a positive integer.
    Now I run into problems, the second bc gives:
    [tex]y(0)=A\sin(n 0)=0[/tex]
    [tex]y'(0)=nA\cos( n 0)=nA[/tex]
    So:
    [tex]y(0)+y'(0) = 0 + nA = 0[/tex]

    Am I on the right track, or can anyone see where I am messing up?
    Thanks.
     
  2. jcsd
  3. Sep 3, 2010 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    No, as a simple counterexample consider [itex]\lambda=\frac{1}{16}[/itex] and [itex]A=-B[/itex]. If [itex]\sqrt{\lambda}[/itex] had to be an integer, then [itex]B[/itex] would have to be zero; but there's no reason to assume [itex]\sqrt{\lambda}[/itex] must be an integer.
     
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