On Lie systems and Kummer-Schwarz equations

TL;DR

This paper uses geometric Lie system theory to show that second- and third-order Kummer--Schwarz equations can be integrated via solutions of a Lie system on SL(2,R), unifying their treatment with related equations.

Contribution

It demonstrates a unified geometric approach to integrate Kummer--Schwarz, Riccati, and Milne--Pinney equations through Lie systems on SL(2,R), simplifying previous methods.

Findings

Integration of Kummer--Schwarz equations reduces to solving a Lie system on SL(2,R)

All related equations can be integrated simultaneously using the same Lie system

The approach generalizes and simplifies previous results in the literature

Abstract

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer--Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,R). This same result can be extended to Riccati, Milne--Pinney and other related equations. We demonstrate that all the above-mentioned equations associated with exactly the same Lie system on SL(2,R) can be integrated simultaneously. This retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.