A Quasi-Lie Schemes Approach to Second-Order Gambier Equations

Abstract

A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.

References in corpus (8)

• Superposition rules, Lie theorem and partial differential equations
• Recent Applications of the Theory of Lie Systems in Ermakov Systems
• Quasi-Lie schemes and Emden--Fowler equations
• Mixed superposition rules and the Riccati hierarchy
• Explicit solutions of the ${\mathfrak{a}}_1$-type Lie-Scheffers system and a general Riccati equation
• Dynamical Studies of Equations from the Gambier Family
• Local and Global Aspects of Lie's Superposition Theorem
• Superposition rules and second-order Riccati equations

Cited by in corpus (1)

• Geometrical and Dynamical Aspects of Nonlinear Higher-Order Riccati Systems