- #1
ShowerHead
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Homework Statement
I have to find the eigenvalues&function of the eqn:
[tex]y''+\lambda y =0[/tex]
With the boundary conditions:
[tex]y(0)+y'(0) = 0[/tex] and [tex]y(\pi) =0[/tex]
Homework Equations
The Attempt at a Solution
I get the general equations, okay, but am having trouble due to the boundary conditions.
Assuming [tex]\lambda>0[/tex], then get the general solution:
[tex]y(x)=A\sin(x\sqrt{\lambda})+B\cos(x\sqrt{\lambda})[/tex]
The best i can do now is that:
[tex]y(\pi)=A\sin(\sqrt{\lambda}\pi)+B\cos(\sqrt{\lambda}\pi)=0[/tex]
Which can only be valid if B = 0 and [tex]\sqrt{\lambda} = n[/tex] where n is a positive integer.
Now I run into problems, the second bc gives:
[tex]y(0)=A\sin(n 0)=0[/tex]
[tex]y'(0)=nA\cos( n 0)=nA[/tex]
So:
[tex]y(0)+y'(0) = 0 + nA = 0[/tex]
Am I on the right track, or can anyone see where I am messing up?
Thanks.