Set u = XT and plug in and I think you should get these two ODEs:
T'' + T' - kT = 0 and X'' - kX = 0, where k is the separation constant.
Solve the equation in X; this should force a condition on k--I think sqrt(k) has to be an integer?
Use that to solve the equation in T.
For each n, u\_n = X\_n T\_n is a solution so an infinite linear combination is a solution.
Use the initial conditions to solve for the coefficients in the infinite sum