7.37, §4 Eigenfunctions

Substitute u=T(y) X(x) into the homogeneous P.D.E. u_xx + u_yy = 0:

TX'' + T'' X = 0

Since the instructor's x-boundary conditions are homogeneous, he has a Sturm-Liouville problem for X:

This was already solved in problem 7.19. Looking back there, substituting ,