A family of quadratic polynomial differential systems with algebraic solutions of arbitrary high degree

R. Ramírez; V. Ramírez

paper

A family of quadratic polynomial differential systems with algebraic solutions of arbitrary high degree

arXiv:1405.2864

Abstract

We show that the algebraic curve $a_{0} (x) (y - r (x)) + p_{2} (x) a^{'} (x) = 0,$ where $r (x)$ and $p_{2} (x)$ are polynomial of degree 1 and 2 respectively and $a_{0} (x)$ is a polynomial solution of the convenient Fucsh's equation, is an invariant curve of the quadratic planar differential system. We study the particular case when $a_{0} (x)$ is an orthogonal polynomials. We prove that that in this case the quadratic differential system is Liouvillian integrable.