Fourier method | eigenfunctions

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.

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!
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.
Last edited:
Yes, I double checked that these are the correct boundary conditions. It is from a PDE course.
In that case I don't know. What physical situation do such conditions come from?

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