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Estimating Eigenvalues from linear ODE

  1. Aug 3, 2017 #1

    joshmccraney

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    1. The problem statement, all variables and given/known data
    Given $$u''(x)+\lambda u = 0\\
    u(-1)=u(1)=0.$$
    If ##\lambda_0## is the lowest eigenvalue, show that ##4 \lambda_0 = \pi^2##.
    2. Relevant equations
    $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}$$ where ##glb## denotes greatest lower bound and ##L## is the Sturm-Louiville operator. I found this equation in the book though I am not sure it is needed.
    3. The attempt at a solution
    We have ##L \equiv -d^2_x##, so $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}\\ = glb\frac{\int_{-1}^1 u''u\,dx}{\int_{-1}^1 u^2\,dx}\\=glb\frac{\int_{-1}^1 u'^2\,dx}{\int_{-1}^1 u^2\,dx}$$
    but from here I'm stuck. I know that last integral should be ##\pi/4## but I'm unsure how to proceed. Perhaps I'm not on the correct track to start? Any ideas?
     
  2. jcsd
  3. Aug 3, 2017 #2

    joshmccraney

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    The solution is to guess ##u=\cos n\pi x/2## and from there it all works out...inspection.
     
  4. Aug 3, 2017 #3

    LCKurtz

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    Or you could just solve the eigenvalue problem. Look at the cases $$
    \lambda = \mu^2 > 0,~\lambda = -\mu^2 < 0, \lambda = 0$$Show only the first case yields non-zero solutions and find the eigenvalues.
     
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