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Consider the following Sturm-Liouville problem: $$X''+\lambda X=0,\quad X'(0) = 0,\, X(\pi) = 0,$$ where $X = X(x)$.

  1. Find all positive eigenvalues and corresponding eigenfunctions of the problem.
  2. Is $\lambda = 0$ an eigenvalue for this problem? If yes, find its eigenfunction. If no, explain why it is not.
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    What have your tried?2012-11-30
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    Have you seen the case $X(0)=0, X(\pi)=0$? Just run the same analysis. Break it into three cases $\lambda>0$, $\lambda=0$ and $\lambda<0$ and see if each are possible.2012-11-30

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