Superposition rules and second-order Riccati equations

TL;DR

This paper extends superposition rules to second-order differential equations, explores their existence, and applies these methods to second-order Riccati equations and other equations of physical interest.

Contribution

It introduces generalizations of superposition rules for second-order equations and investigates their existence and connections with Lie systems and quasi-Lie schemes.

Findings

Generalized superposition rules for second-order equations

Existence results for these generalizations

Applications to second-order Riccati equations

Abstract

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose several generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.