The role of algebraic solutions in planar polynomial differential systems

Abstract

AbstractWe study a planar polynomial differential system, given by$\dot{x}=P(x,y)$, $\dot{y}=Q(x,y)$. We consider a function$I(x,y)=\exp\!\{h_2(x) A_1(x,y) \diagup A_0(x,y) \}$ $ h_1(x)\prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}$, wheregi(x) are algebraic functions of$x$, $A_1(x,y)=\prod_{k=1}^r (y-a_k(x))$, $A_0(x,y)=\prod_{j=1}^s (y-\tilde{g}_j(x))$withak(x) and$\tilde{g}_j(x)$algebraic functions,A0(x,y) andA1(x,y) do not share any common factor,h2(x) is a rational function,h(x) andh1(x) are functions ofxwith a rational logarithmic derivative and$\alpha_i \in \mathbb{C}$. We show that ifI(x,y) is a first integral or an integrating factor, thenI(x,y) is a Darboux function. A Darboux function is a function of the form$f_1^{\lambda_1} \cdots f_p^{\lambda_p} \exp\{h/f_0\}$, wherefiandhare polynomials in$\mathbb{C}[x,y]$and the λi's are complex numbers. In order to prove this result, we show that ifg(x) is an algebraic particular solution, that is, if there exists an irreducible polynomialf(x,y) such thatf(x,g(x)) ≡ 0, thenf(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 formI(x,y) such as the structure of their cofactor.Moreover, we considerA0(x,y),A1(x,y) andh2(x) as before and a function of the form$\Phi(x,y):= \exp \{h_2(x)\, A_1(x,y)/A_0 (x,y) \}$. 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$R=h_2 A_1/A_0 + B_1/B_0$and withB1/B0a function of the same form ash2A1/A0. Hence, exp {R(x,y)} factorizes as the product Φ(x,y) Ψ(x,y), for Ψ(x,y): = exp {B1/B0.