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

TL;DR

This paper constructs a family of quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree, demonstrating their Liouvillian integrability when associated with orthogonal polynomials.

Contribution

It introduces a new class of quadratic systems with high-degree algebraic solutions and proves their Liouvillian integrability in special cases.

Findings

Invariant algebraic curves of arbitrary degree are constructed.

Quadratic systems with orthogonal polynomial solutions are Liouvillian integrable.

The algebraic solutions are characterized by Fucsh's equation.

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.