First integrals for the damped Helmholtz oscillator

Abstract

Recently, the modified Prelle–Singer method for finding general solutions of second-order nonlinear ordinary differential equations has attracted considerable attention. Many researchers used this method to derive the first integrals of various dynamical systems. In this article, we are concerned with the first integrals of the damped Helmholtz oscillator under certain parametric conditions. Our analysis indicates that there exist some errors on the first integrals of the damped Helmholtz oscillator and the Duffing–van der Pol oscillator in the literature. We clarify the errors, and instead give a refined result in a simple and straightforward manner with much less calculations. Finally, two independent first integrals of the Helmholtz oscillator under different parametric conditions are obtained by the Lie symmetry method, and a class of exact solutions in terms of the hyperbolic and the Jacobian elliptic functions is presented through these two first integrals.