Solutions of an extended Duffing-van der Pol equation with variable coefficients

TL;DR

This paper derives exact solutions for a nonlinear Duffing-van der Pol oscillator with variable coefficients using two different analytical methods, and explores its Lagrangian structure and specific examples.

Contribution

It introduces two novel analytical approaches to solve the variable coefficient Duffing-van der Pol equation and derives its Lagrangian formalism.

Findings

Exact solutions obtained via factorization and Painlevé analysis

Lagrangian formalism established for the system

Examples illustrating specific solutions for particular coefficients

Abstract

In this work, exact solutions of the nonlinear cubic-quintic Duffing-van der Pol oscillator with variable coefficients are obtained. Two approaches have been applied, one based on the factorization method combined with the Field Method, and a second one relying on Painlev\'e analysis. Both procedures allow us to find the same exact solutions to the problem. The Lagrangian formalism for this system is also derived. Moreover, some examples for particular choices of the time-dependent coefficients, and their corresponding general and particular exact solutions are presented.