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Eigenvalues, Eigenfunctions and Sturm-Liouville Problems

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1. y'' + k*y = 0 BC: y'(0) = 0 y'(L) = 0

2. y'' + k*y = 0 BC: y(0) = y(&#61552;) y'(0) = y'(&#61552;)

3. y'' + k*y = 0 BC: y(0) = 0 y(&#61552;) +2*y'(&#61552;) = 0

4. y'' + 2*y' + (1+k)*y = 0 BC: y(0) = y(1) =0

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Eigenvalues, Eigenfunctions and Sturm-Liouville Problems are investigated. The solution is detailed and well presented.

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1. y'' + k*y = 0 BC: y'(0) = 0 y'(L) = 0

It's a Sturm-Liouville problem. We have 3 possible cases here:

(1) k < 0. Let's write , where . Then the equation becomes and its general solution is . If we use the first boundary condition, y'(0) = 0, we get . If we use the second condition y'(L) = 0, we get , and since unless x = 0, we infer that . Thus there are no nonzero solutions in this case.

(2) k = 0. The general solution in this case is and boundary conditions would make , hence leaving us with as a solution.

(3) k > 0. Then we can write , where . Then the equation becomes and its general solution is . If we use the first boundary condition, y'(0) = 0, we get . If we use the second condition y'(L) = 0, we get , and since we are seeking nonzero solutions, we take . Thus we must have . Thus , consequently - eigenvalues and corresponding eigenfunctions ...

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