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EngageEngage
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Homework Statement
Find the eigenvalues and the eigenfunctions of the Sturm-Liouville problem
[tex] \frac{d^{2}u}{dx^{2}}=\lambda u [/tex]
[tex] 0<x<L[/tex]
[tex]\frac{du}{dx}(0) = 0[/tex]
[tex]u(L) = 0[/tex]
The Attempt at a Solution
characteristic polynomial:
[tex] p^{2}=+-\lambda[/tex]
[tex]u = Ae^{\sqrt{\lambda}x}+Be^{-\sqrt{\lambda}x}[/tex]
[tex]u = Ccosh(\sqrt{\lambda}x)+Dsinh(-\sqrt{\lambda}x)[/tex]
Now, i try to solve the boundaries:
[tex]
\frac{du}{dx}(0)=-D\sqrt{\lambda}cosh(-\sqrt{\lambda}x)=0
[/tex] ... I am confused now because cosh doesn't have a root unless its translated. Can anyone help me out with this please?