The role of algebraic solutions in planar polynomial differential systems
The role of algebraic solutions in planar polynomial differential systems
Abstract We 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}$ , where g i ( x ) are algebraic functions of $x$, $A_1(x,y)=\prod_{k=1}^r (y-a_k(x))$, $A_0(x,y)=\prod_{j=1}^s (y-\tilde{g}_j(x))$ with a k ( x ) and $\tilde{g}_j(x)$ algebraic …