Homework Help: Sturm-Liouville problem

1. The problem statement, all variables and given/known data
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]

3. 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?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

 

O yeah, thanks. not sure how i managed to screw that up. then I get
[tex] u = Ccosh(\sqrt{\lambda x})[/tex]
[tex] u(L) = 0 = Ccosh(\sqrt{\lambda L})[/tex]
But cosh has no root here so, i get u = 0, by setting C = 0.
I just realized it: i probably have to do this with lambda>0, <0 and = 0, is that righT?

 

so when: [tex] \lambda = 0[/tex]
[tex]
u = C(1)+D(0);;;
u = C
[/tex]
and then i get u = 0 again.

now when [tex] \lambda <0[/tex]
[tex] u = Ae^{i\sqrt{|\lambda|}x}+Be^{-i\sqrt{|\lambda|}x}[/tex]
[tex] u = Ecos(\sqrt{\lambda}x)+Fsin(\sqrt{\lambda}x)[/tex]
[tex] u'(0)=0 when F = 0[/tex]
[tex] u = Ecos(\sqrt{\lambda}x)[/tex]
[tex] u(L)=0=Ecos(\sqrt{\lambda}L)[/tex]
[tex] \lambda = (\frac{(n-\frac{1}{2})\pi}{L})^{2}, n = 1,2,3...[/tex]
[tex] u = cos(\frac{(n-\frac{1}{2})\pi x}{L})[/tex]

 

is that right?

 

messed that last part up, but i got it now. Thanks!