Periodic Solutions of a Nonlinear Differential Equation

Periodic solutions of the nonlinear differential equation ẍ + f(x) = 0 are investigated. The qualitative characteristics of the solution are explored and several theorems about the basic nature of the solution are presented. The solution is then obtained in terms of a Fourier series and numerical calculations are carried out to fifth-order accuracy. The dependence of the frequency of the solution on the amplitude is studied. As an example, the solution is applied to the problem of obtaining the orbit of a charged particle moving in a radial electric field.