In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction?
That is to say, if I end up with an eigenvalue of zero, am I going to add it to the sum that I get from solving for the other eigenvalues? If so, am I going to multiply it by my T(t) first? Something like:
$X_0(x)T(t) + \sum a_nT(t)X_n(x)$ or do we just list 0 as one of our eigenvalues, and then we don't make any changes to the way the function is written out?