Integrability of planar polynomial differential systems through linear differential equations

Details Integrability of planar polynomial differential systems through linear differential equations Integrability of planar polynomial differential systems through linear differential equations

2005-06-02

arxiv.org/abs/math/0506035v1

Mathematics

Dynamical Systems

24 pages, no figures

Scientific paper

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.

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