%0 Journal Article
%T A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
%A Jos¨¦ F. Cari£¿ena
%A Partha Guha
%A Javier de Lucas
%J Symmetry, Integrability and Geometry : Methods and Applications
%D 2013
%I National Academy of Science of Ukraine
%X 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.
%K Lie system
%K Kummer-Schwarz equation
%K Milne-Pinney equation
%K quasi-Lie scheme
%K quasi-Lie system
%K second-order Gambier equation
%K second-order Riccati equation
%K superposition rule
%U http://dx.doi.org/10.3842/SIGMA.2013.026