Exact solutions of the Li\'enard and generalized Li\'enard type ordinary non-linear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator

TL;DR

This paper introduces a novel method inspired by quantum mechanics to derive exact solutions for nonlinear Lie9nard-type differential equations by deforming the phase space of the harmonic oscillator, with applications in physics.

Contribution

It presents a new phase space deformation technique to obtain explicit solutions of nonlinear differential equations related to harmonic oscillators.

Findings

Derived exact solutions for Lie9nard and generalized Lie9nard equations.

Connected solutions to physical systems like wave propagation and beam vibrations.

Generalized method applicable to broader classes of nonlinear equations.

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.