Integrability of planar polynomial differential systems through linear differential equations
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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.
Is part ofRocky Mountain Journal of Mathematics, 2006, vol. 36, núm 2, p. 457-485
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Giacomini, Héctor; Giné, Jaume; Grau Montaña, Maite (Cambridge University Press, 2007)We study a planar polynomial differential system, given by . We consider a function , where gi(x) are algebraic functions of with ak(x) and algebraic functions, A0(x,y) and A1(x,y) do not share any common factor, h2(x) is ...
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