Integrability of planar polynomial differential systems through linear differential equations

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Issue date
2006Suggested citation
Giacomini, Héctor;
Giné, Jaume;
Grau Montaña, Maite;
.
(2006)
.
Integrability of planar polynomial differential systems through linear differential equations.
Rocky Mountain Journal of Mathematics, 2006, vol. 36, núm 2, p. 457-485.
https://doi.org/10.1216/rmjm/1181069462.
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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 family.
We give an analogous method for constructing rational
equations but by means of a linear differential equation of
first order.