A generalization of the S-function method applied to a Duffing-Van der Pol forced oscillator

TL;DR

This paper extends the S-function method to handle elementary function-based second order ODEs and applies it to a Duffing-Van der Pol oscillator to find a broad class of first integrals.

Contribution

The paper generalizes the S-function method to include elementary functions and demonstrates its effectiveness on a complex forced oscillator.

Findings

Successfully generalized the S-function method for elementary 2ODEs.

Derived an entire class of first integrals for the Duffing-Van der Pol oscillator.

Enhanced analytical tools for solving resistant differential equations.

Abstract

In [1,2] we have developed a method (we call it the S-function method) that is successful in treating certain classes of rational second order ordinary differential equations (rational 2ODEs) that are particularly `resistant' to canonical Lie methods and to Darbouxian approaches. In this present paper, we generalize the S-function method making it capable of dealing with a class of elementary 2ODEs presenting elementary functions. Then, we apply this method to a Duffing-Van der Pol forced oscillator, obtaining an entire class of first integrals.