The approximate solution of non-linear differential equations is studied to second-order using the method of harmonic balance with generalized Fourier series and Jacobian elliptic functions. As an interesting use of the series, very good analytic approximations to the limit cycles of Liénard's ordinary differential equation, [Xdot] + g(X) = f(X)[Xdot] are presented. In the generalized van der Pol equation with f(X)= ϵ(1 − X²) and g(X) = AX + 2BX₃ a very good second-order approximation is given that depends on the values of A/B and ϵ.