European Journal of Applied Mathematics

Table of Contents - 2006 - Volume 17, Issue 02  


Papers

An algorithmic method to determine integrability for polynomial planar vector fields

HECTOR GIACOMINI a1 and JAUME GINÉ a2
a1 Laboratoire de Mathématique et Physique Théorique, CNRS (UMR 6083), Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 Tours, France email: Hector.Giacomini@phys.univ-tours.fr
a2 Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69. 25001 Lleida, Spain email: gine@eps.udl.es

Article author query
giacomini h   [Google Scholar] 
gine j   [Google Scholar] 
 

Abstract

In this paper we study some aspects of the integrability problem for polynomial vector fields $\dot{x}=P(x,y)$, $\skew1\dot{y}=Q(x,y)$. We analyze the possible existence of first integrals of the form $I(x,y)=e^{ h_1(x) \prod_{k=1}^r (y-a_k(x))/ \prod_{j=1}^s(y-f_j(x))} h_2(x)$ $\prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}$, where $g_i(x)$ and $f_j(x)$ are unknown particular solutions of $dy/dx=Q(x,y)/P(x,y)$, $\alpha_i \in \mathbb{C}$ are unknown constants, and $a_k(x)$, $h_1(x)$ and $h_2(x)$ are unknown functions. We give an algorithmic method to determine if the polynomial vector field has a first integral of the form above described. In the case when some of the particular solutions remain arbitrary and the other ones are explicitly determined or are functionally related to the arbitrary particular solutions, we will obtain a generalized nonlinear superposition principle, see [6]. In the case when all the particular solutions $g_i(x)$ and $f_j(x)$ are determined, they are algebraic functions and our algorithm gives an alternative method for determining such type of solutions.

(Published Online July 19 2006)
(Received September 22 2004)
(Revised August 4 2005)