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Homework Help: Help finding a general solution for an eigenvalue problem

  1. Sep 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Hey, guys. I'm having trouble finding the general solution to a second order, homogeneous ODE. It is the first step to solving an eigenvalue problem and my professor is about as much help as a hole in the head. I've tried multiple "guesses" and have combed various resources looking for general solutions and have come up empty. Can someone help out, please?

    2. Relevant equations
    X''(x) + X'(x) - λX(x) = 0
    X(0) = 0, X(a) = 0

    3. The attempt at a solution
    I cannot even solve the characteristic equation, m^2 + m - λ = 0 because you end up getting √1+4λ under the radical in the quadratic equation. In summary, I'm stuck, haha.
    Last edited: Sep 28, 2012
  2. jcsd
  3. Sep 28, 2012 #2


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    Science Advisor

    What's wrong with that? [itex]\lambda[/itex] is just a number. There is no reason in the world why it could not be "under the radical".

    By the way, it is not [itex]\sqrt{1- \lambda^2}[/itex]. By the quadratic formula, the solution to [itex]m^2+ m- \lambda= 0[/itex] is
    [tex]\frac{-1\pm\sqrt{1^2- 4(1)(-\lambda)}}{2(1)}= \frac{-1\pm\sqrt{1+ 4\lambda}}{2}[/tex]

    Now, you would want to consider the cases [itex]1+ 4\lambda> 0[/itex], [itex]1+ 4\lambda= 0[/itex], and [itex]1+ 4\lambda< 0[/itex] separately.
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