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ODE/PDE- eighenvalues question

  1. Oct 4, 2010 #1
    1. The problem statement, all variables and given/known data
    X''+[tex]\lambda[/tex]*X=0 0<x<2pi

    X(0)=X(2pi) and X'(0)=X'(2pi)

    2. Relevant equations

    3. The attempt at a solution
    My only problem is the case when [tex]\lambda[/tex]=0
    in such case the general solution is X(x)= C1+C2*x

    Now, after applying the BCs, this is what I get:



    now, what should C2 be ? zero or any value ?
    if C2 is zero, we have the trivial solution only. However, if C2 is any number, we have a nontrivial solution.

    how can I solve this question?

  2. jcsd
  3. Oct 4, 2010 #2


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    After applying the first BC, I get C1=C1+C2*2*pi, not C1=C2*2*pi.
  4. Oct 4, 2010 #3


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    This should be

    [tex]C_1 = C_1 + 2\pi C_2.[/tex]

    How does the solution with [tex]\lambda\neq 0[/tex] behave in the limit as [tex]\lambda \rightarrow 0[/tex]? Is this consistent with your solution for [tex]\lambda = 0[/tex]?
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