Quasi-Lie schemes: theory and applications

TL;DR

This paper introduces Quasi-Lie schemes, a geometric approach to solving non-linear first-order ODEs, providing both solutions and explanations for existing methods.

Contribution

It develops the theory of Quasi-Lie schemes and demonstrates their application to solving and understanding non-linear first-order differential equations.

Findings

Provides a new geometric framework for Lie systems

Enables solving certain non-linear ODEs more effectively

Offers explanations for existing ad hoc solution methods

Abstract

A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to solve some of these equations, but also gives a geometrical explanations for some, already known, ad hoc methods of dealing with such problems.