Superposition rules and second-order differential equations

TL;DR

This paper extends superposition rules from first-order to second-order differential equations, enabling the application of Lie and quasi-Lie theories to analyze complex equations in physics and mathematics.

Contribution

It introduces new generalizations of superposition rules for second-order differential equations and demonstrates their application using examples from literature.

Findings

Derived superposition rules for several second-order equations

Applied Lie and quasi-Lie theories to analyze these equations

Provided new tools for solving second-order differential equations

Abstract

The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply the theories of Lie and quasi-Lie systems to analyse second-order differential equations. In order to illustrate our results, several second-order differential equations appearing in the physics and mathematical literature are analysed and some superposition rules for these equations are derived by means of our methods.