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!