Sturm-Liouville problem

  • #1
208
0

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?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
[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?
Hi EngageEngage!

Isn't it just D = 0? :smile:
 
  • #3
208
0
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?
 
  • #4
208
0
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]
 
Last edited:
  • #6
208
0
messed that last part up, but i got it now. Thanks!
 
  • #7
tiny-tim
Science Advisor
Homework Helper
25,832
251
Hi EngageEngage! :smile:

Yes, that looks fine!

(though personally I don't like minuses, so I'd write it:

[tex] u = cos\left(\frac{(n+\frac{1}{2})\pi x}{L}\right)[/tex] )

(note the \left( and \right) in the LaTeX! :smile: )
 

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