On Lie systems and KummerSchwarz equations
Abstract
A Lie system is a system of firstorder 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 thirdorder KummerSchwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,{R}). This same result can be extended to Riccati, MilnePinney, and to the here defined generalised KummerSchwarz equations, which include several types of KummerSchwarz equations as particular cases. We demonstrate that all the abovementioned equations related to the same Lie system on SL(2,{R}) can be integrated simultaneously, which 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 March 2013
 DOI:
 10.1063/1.4794280
 arXiv:
 arXiv:1212.5779
 Bibcode:
 2013JMP....54c3505D
 Keywords:

 differential equations;
 geometry;
 integration;
 Lie groups;
 02.30.f;
 02.40.k;
 02.60.Jh;
 02.20.Sv;
 Function theory analysis;
 Geometry differential geometry and topology;
 Numerical differentiation and integration;
 Lie algebras of Lie groups;
 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 34A26
 EPrint:
 29 pages. A relevant error and several typos corrected