Integrability of Planar Polynomial Differential Systems through Linear Differential Equations
Résumé
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