What does the choice of c-ak for the separation constant for the time ODE imply for the choice of c, the separation constant for the space ODE? Solve the space ODE to find two linearly independent solutions (actually, we get an infinity of these two for each possible value of k). Thus write down a full basis of separable solutions to our original problem. Let's solve a Dirichlet problem: First we will consider fixed boundary conditions for the ends of the bar, u(0, t) = u(L,t) = 0. We will also assume we know the temperature profile at time t = 0. u(t,0) = f(x).