1. The problem statement, all variables and given/known data Solve the diffusion equation with the boundary conditions v(0,t)=0 for t > 0 and v(x,0) = c for t=0. The method should be separation of variables. 2. Relevant equations The separation of variables method. 3. The attempt at a solution Attempting a solution of the form XT leads you to an exponential for T and a sinusoidal for X: X = Asin(kx) + Bcos(kx) where -k^2 was the constant used for solving the two separated differential equations. However. My solution manuals writes the constants A and B as a continious function of the parameter k, and I don't understand why. Why do the constants, which are chosen from the boundary conditions have anything to do with k? And going further the full solution is then written as a fourier integral from 0 to ∞ of XTB(k)dk Where on earth does this come from? Note that A(k)=0 from the boundary conditions. Can someone try to explain why you most impose a continuous superposition like the above to get the general solution?