In a previous paper (Ottoy 1979) a perturbation method for second order non-linear differential equations was explained. In this paper an application of the method is treated for the differential equation : $. For this equation, which is perturbed by a Van der Pol damping factor, the non-perturbed solutions are periodic and can be expressed with the elliptic functions of Jacobi. Not only excellent solution accuracy, but also an approximation of the equation of the limit-cycle is obtained.