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Eigenvalue solutions of the transcendental equation

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

    Given the Sturm-Liouville system:

    y'' + λy = 0 , y(0) - y'(0) = 0 , y(1) + y'(1) = 0

    Show using the Rayleigh Quotient that the eigenvalues are positive.

    Show that these eigenvalues are given as the solutions of the transcendental equation:

    tan ( √λ ) = ( 2 √λ) / ( λ - 1 )

    2. Relevant equations

    Using the Rayleigh Quotient, I found the eigenvalues to be:

    λn = ( yn(1)2 + yn(0)2 + ∫01 yn'2dx ) / ∫01yn2dx

    Therefore, as the eigenfunction yn cannot be zero, the eigenvalues λn are positive for all n.

    3. The attempt at a solution

    Then I have no idea how to show that these eigenvalues are solutions to the transcendental equation. I've tried putting the eigenvalues into the equation and can't figure out to simplify or to show that they solve the equation?

    Thanks for any help.
     
  2. jcsd
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