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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] ... im confused now because cosh doesn't have a root unless its translated. Can anyone help me out with this please?