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Eigenfunction 2nd order DE problem

  1. Oct 16, 2011 #1
    1. The problem statement, all variables and given/known data

    For the following equation

    [itex]y" - λy=0[/itex]

    find the values of λ which produce a non-trivial solution on the interval 0 <= x <= a

    The given initial conditions are

    y(0) = 0
    y(a) = 0

    2. Relevant equations



    3. The attempt at a solution

    see attached pdfs

    My problem is im not being able to produce a non-trivial solution for all three cases. It's further complicated by the fact that the tutor hasn't provided any answer to the problem

    I just want to make sure my understanding and approach to the problem is reasonable. If anyone has a good answer to the problem, please let me know
     

    Attached Files:

  2. jcsd
  3. Oct 17, 2011 #2

    LCKurtz

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    Science Advisor
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    Gold Member

    It looks pretty much OK right up to the end. The equation is y''-λy = 0, y(0)=y(a) = 0.

    You are correct that λ > 0 and λ = 0 yield no nontrivial solutions. In the case λ < 0 you can write λ = -k2 so the equation as

    y'' + k2y = 0

    Then at the end when you get to Bsin(ka) = 0, to avoid B = 0 you must have
    ka = nπ so k = nπ/a and λn = -(nπ/a)2. The eigenvalues are always real in this type of problem.
     
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