Polynomial and rational first integrals for planarquasi--homogeneous polynomial differential systems

Jaume Giné; Maite Grau; Jaume Llibre

doi:10.3934/dcds.2013.33.4531

Article Contents

2013, Volume 33, Issue 10: 4531-4547. Doi: 10.3934/dcds.2013.33.4531

This issue Previous Article Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations Next Article A Liouville theorem of degenerate elliptic equation and its application

Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems

1.
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia

Received: November 2012

Revised: January 2013

Published: October 2013

Abstract / Introduction Related Papers Cited by

Abstract

In this paper we find necessary and sufficient conditions in order that a planar quasi--homogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasi--homogeneous polynomial differential system can be transformed into a differential system of the form $\dot{u} \, = \, u f(v)$, $\dot{v} \, = \, g(v)$ with $f(v)$ and $g(v)$ polynomials, and vice versa.

Keywords:

Quasi--homogeneous polynomial differential equations,
integrability problem,
polynomial first integral,
rational first integral.

Mathematics Subject Classification: Primary: 34C05, 34A34, 34C20.

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