Here are the boundary conditions for this example

and finally initial conditions are

If X(0)T(t) = 0 then either X(0) = 0 or T(t) = 0. If T(t) = 0 then u(x, t) = 0 which is a trivial solution and it is not of interest; therefore X(0) = 0. Similarly one can conclude from X(1)T(t) = 0 that X(1) = 0.