The role of algebraic solutions in planar polynomial differential systems
View/Open
Issue date
2007Suggested citation
Giacomini, Héctor;
Giné, Jaume;
Grau Montaña, Maite;
.
(2007)
.
The role of algebraic solutions in planar polynomial differential systems.
Mathematical Proceedings of the Cambridge Philosophical Society, 2007, vol. 143, núm. 2, p. 487508.
https://doi.org/10.1017/S0305004107000497.
Metadata
Show full item recordAbstract
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 a rational function, h(x) and h1(x) are functions of x with a rational logarithmic derivative and . We show that if I(x,y) is a first integral or an integrating factor, then I(x,y) is a Darboux function. A Darboux function is a function of the form , where fi and h are polynomials in and the λi's are complex numbers. In order to prove this result, we show that if g(x) is an algebraic particular solution, that is, if there exists an irreducible polynomial f(x,y) such that f(x,g(x)) ≡ 0, then f(x,y) = 0 is an invariant algebraic curve of the system. In relation with this fact, we give some characteristics related to particular solutions and functions of the form I(x,y) such as the structure of their cofactor.
Moreover, we consider A0(x,y), A1(x,y) and h2(x) as before and a function of the form . We show that if the derivative of Φ(x,y) with respect to the flow is well defined over {(x,y): A0(x,y) = 0} then Φ(x,y) gives rise to an exponential factor. This exponential factor has the form exp {R(x,y)} where and with B1/B0 a function of the same form as h2A1/A0. Hence, exp {R(x,y)} factorizes as the product Φ(x,y) Ψ(x,y), for Ψ(x,y): = exp {B1/B0.
Is part of
Mathematical Proceedings of the Cambridge Philosophical Society, 2007, vol. 143, núm. 2, p. 487508European research projects
Collections
Related items
Showing items related by title, author, creator and subject.

Integrability of planar polynomial differential systems through linear differential equations
Giacomini, Héctor; Giné, Jaume; Grau Montaña, Maite (Rocky Mountain Mathematics Consortium, 2006)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 ... 
Classical planar algebraic curves realizable by quadratic polynomial differential systems
García, I. A. (Isaac A.); Llibre, Jaume (World Scientific Publishing, 20171115)In this paper we show planar quadratic polynomial differentialsystems exhibiting as solutions some famous planar invariant algebraic curves. Also we put particular attention to the Darboux integrability of these differential ... 
Polynomial and rational first integrals for planar quasihomogeneous polynomial differential systems
Giné, Jaume; Grau Montaña, Maite; Llibre, Jaume (American Institute of Mathematical Sciences, 201310)In this paper we find necessary and sufficient conditions in order that a planar quasihomogeneous polynomial differential system has a polynomial or a rational first integral. We also prove that any planar quasihomogeneous ...