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Differential Equations - Eigenvalues and Eigenfunctions

  1. Apr 30, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the eigenvalues and the eigenfunctions for
    x^2y"+2xy'+λy = 0 y(1) = 0, y(e^2) = 0


    2. Relevant equations

    See problem

    3. The attempt at a solution
    My book has one paragraph on this that does not help me. I tried using an auxiliary equation and solving for lambda.
    I get (m+√λ)^2 a real number with duplicity two. I don't know what to do with this or if this is even the right first step. The SINGLE paragraph on this doesn't explain anything (it's practically a side note)

    I would appreciate some guidance. There are two problems like this on the take home quiz and we didn't go over it in class so I am very lost and confused.
    Also I've been pounding my head over this with my directionless fury. Help subside my hate!

    Thanks for your time everyone!
     
  2. jcsd
  3. Apr 30, 2012 #2

    HallsofIvy

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    You mean "multiplicity" 2 ("duplicity" has a completly different meaning!). And, while you say you "get (m+√λ)^2" you don't say what that is supposed to be- particularly since there is no "m" in the problem. Please show how you got that.

    (I'm not sure what you mean by "m" but I get nothing like that. It looks to me like [itex]\lambda[/itex] will have to be larger than 1/4 in order to get a non-trivial function that is 0 for two different x values.)
     
    Last edited: Apr 30, 2012
  4. Apr 30, 2012 #3
    I got it from changing the DE into
    m^2 + 2m + λ
    I did the "m" thing because in the previous chapters to solve some equations I would solve the associated homogeneous DE and use the value of m to determine the structure of the general solution.
    It's the only step I have made and I feel like even that is wrong. I don't know what to do with the initial values or anything...
    Thanks for clearing up the duplicity multiplicity thing, you're right :)

    My problem is a complete lack of understanding of what I am supposed to do, and my textbook is not helping haha...
     
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