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Math Help - how to show this DE has infinitely many eigenvalues

  1. #1
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    how to show this DE has infinitely many eigenvalues

     y''+\lambda y=0 \ \ \ \ \ \ y(0)=0, \ \ y(1)+y'(1)=0

    has infinitely many eigenvalues  \lambda_{1}<\lambda_{2}<...

    and indicate the behaviour of lambda as n goes to infinity.
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  2. #2
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    You really ought not to need much of a hint in order to do this. You should recognise  y''+\lambda y=0 as a simple harmonic motion equation (assuming that \lambda>0), with solution y=A\cos\mu x + B\sin\mu x, where \mu = \pm\sqrt\lambda. The initial conditions will give you a condition on \mu, of the form \tan\mu=-\mu. This is not an equation that you can solve explicitly, but by drawing a graph of the tan function, you can see that there is a doubly-infinite family of solutions, which for large values of |\mu| will be close to odd multiples of \pi/2.
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