Algebraic invariant curves and first integrals for Riccati polynomial differential systems

Llibre, Jaume; Valls, Clàudia

doi:10.1090/S0002-9939-2014-12085-5

Algebraic invariant curves and first integrals for Riccati polynomial differential systems
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by Jaume Llibre and Clàudia Valls PDF

Proc. Amer. Math. Soc. 142 (2014), 3533-3543 Request permission

Abstract:

We study the algebraic invariant curves and first integrals for the Riccati polynomial differential systems of the form $x’=1$, $y’ =a(x) y^2 +b(x)y +c(x)$, where $a(x)$, $b(x)$ and $c(x)$ are polynomials. We characterize them when $c(x)=\kappa (b(x)-\kappa a(x))$ for some $\kappa \in \mathbb {C}$. We conjecture that algebraic invariant curves and first integrals for these Riccati polynomial differential systems only exist if $c(x)=\kappa (b(x)-\kappa a(x))$ for some $\kappa \in \mathbb {C}$.

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