Exact solutions of the Liénard- and generalized Liénard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator

Abstract

We investigate the connection between the linear harmonic oscillator equation and some classes of second order nonlinear ordinary differential equations of Li\'enard and generalized Li\'enard type, which physically describe important oscillator systems. By using a method inspired by quantum mechanics, and which consist on the deformation of the phase space coordinates of the harmonic oscillator, we generalize the equation of motion of the classical linear harmonic oscillator to several classes of strongly non-linear differential equations. The first integrals, and a number of exact solutions of the corresponding equations are explicitly obtained. The devised method can be further generalized to derive explicit general solutions of nonlinear second order differential equations unrelated to the harmonic oscillator. Applications of the obtained results for the study of the travelling wave solutions of the reaction-convection-diffusion equations, and of the large amplitude free vibrations of a uniform cantilever beam are also presented.