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How to find solutions to a Legendre equation?

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the first three positive values of [itex]\lambda[/itex] for which the problem:

    [itex](1-x^2)y^n-2xy'+\lambda y = 0, \ y(0)=0, \ y(x)[/itex] & [itex]y'(x)[/itex] bounded on [itex][-1, 1][/itex]

    has nontrivial solutions.

    2. Relevant equations

    When n is even:

    [itex]y_1(x) = 1 - \frac{n(n+1)}{2!}x^2 + \frac{(n-2)(n(n+1)(n+3)}{4!} - ...[/itex]

    When n is odd:

    [itex]y_2(x) = x - \frac{(n-1)(n+2)}{3!} + \frac{(n-3)(n-1)(n+2)(n+4)}{5!} - ...[/itex]

    3. The attempt at a solution

    I was able to do part of the problem myself. By plugging in zero into both of the above equations and then comparing to initial conditions, I was able to determine that n is odd. However, I don't know what to do next.

    My textbook does not have *any* examples for this type of problem and I haven't been able to find anything I understand online. I looked the problem up on Chegg and found it here:


    Steps 1-2 on there are basically what I did, but then they jump to the answer in step 3 with zero explanation as to how they got there.

    I don't want to just copy what they did with no understanding of how they got there, I want to actually understand it.. Can anyone explain what I need to do next to get the solutions?

    Thanks :)
  2. jcsd
  3. Dec 9, 2013 #2


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    The usual approach is to find a series solution. The requirement that y and y' remain bounded requires that ##\lambda## take on only certain values.
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