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Article Contents

# Analytic integrability of a class of planar polynomial differential systems

• In this paper we find necessary and sufficient conditions in order that the differential systems of the form $\dot x = x f(y)$, $\dot y =g(y)$, with $f$ and $g$ polynomials, have a first integral which is analytic in the variable $x$ and meromorphic in the variable $y$. We also characterize their analytic first integrals in both variables $x$ and $y$.
These polynomial differential systems are important because after a convenient change of variables they contain all quasi--homogeneous polynomial differential systems in $\mathbb{R}^2$.
Mathematics Subject Classification: Primary: 34A34, 34C14.

 Citation:

•  [1] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete and Continuous Dynamical Systems, Series A, 33 (2013), 4531-4547.doi: 10.3934/dcds.2013.33.4531. [2] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover Publications, Inc., New York, 1994. [3] J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.doi: 10.1088/0951-7715/15/4/313.