Is sin[(n+1/2)x] also an eigenfunction for this problem?

[kingwinner]

Use separation of variables/Fourier method to solve
u_t - 4u_xx = 0, -pi<x<pi, t>0
u(-pi,t) = -u(pi,t), u_x(-pi,t) = -u_x(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.

 

[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?