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Integrability of planar polynomial differential systems through linear differential equations

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Issue date
2006
Author
Giacomini, Héctor
Giné, Jaume
Grau Montaña, Maite
Suggested 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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Abstract
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.
URI
http://hdl.handle.net/10459.1/57788
DOI
https://doi.org/10.1216/rmjm/1181069462
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Rocky Mountain Journal of Mathematics, 2006, vol. 36, núm 2, p. 457-485
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