# Fourier method | eigenfunctions

#### kingwinner

Use separation of variables/Fourier method to solve
ut - 4uxx = 0, -pi<x<pi, t>0
u(-pi,t) = -u(pi,t), ux(-pi,t) = -ux(pi,t), t>0.
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What I got is that (n+1/2)2 are eigenvalues (n=0,1,2,3,...) and cos[(n+1/2)x)] is an eigenfunction.
Instead of two sets of eigenvalues, there is only one set. I cannot find another set of eigenvalues.
My question is: is sin[(n+1/2)x)] also an eigenfunction for this problem? Why or why not?

Thanks for any help!

#### Phyisab****

Are you sure those are the correct boundary conditions? What problem led you to those boundary conditions? They seem to imply only that u and du/dx are odd functions of x, not that u is smooth at pi.

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#### kingwinner

Yes, I double checked that these are the correct boundary conditions. It is from a PDE course.

#### Phyisab****

In that case I don't know. What physical situation do such conditions come from?

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