DFT approximation of a function with a finite sum of harmonics gets better as the number of grid points, and therefore the number of harmonics, is increased. In the limit, as the number of grid points tends to infinity, the finite sum becomes a Fourier series which represents the function exactly. Fourier series representation is thus DFT approximation brought to its logical conclusion. Previously, we used DFT to approximate the solution of a first order scalar linear ODE with constant coefficients. We will now show that the exact solution can be easily found as a Fourier series. Finding Fourier series solutions of higher order ODE and matrix-vector systems is just as simple.