"Verify that, for any C¹ function f(x), u(x, t) = f(x - ct) is a solution of the PDE u_t + c u_x = 0, where c is a constant and u_t and u_x are partial derivatives." I managed to get the solution for this and a similar problem by showing that the new variable (x - ct in this case) satisfies the PDE, but why does doing that work? Is it because if x-ct is a solution so is any function of it? That just doesn't sound right since you'd need at least two linearly independent solutions to make that kind of generalization. Or maybe I'm just an idiot. :) Mucho thanks if anyone can point out what basic thing I'm missing.