Solution of the system of two coupled first-order ODEs with second-degree polynomial right-hand sides

TL;DR

This paper provides explicit solutions for a subclass of autonomous systems of two coupled first-order ODEs with quadratic right-hand sides, including formulas relating coefficients and parameters, and identifies special cases with periodic solutions.

Contribution

It explicitly solves the initial-values problem for a subclass of quadratic coupled ODEs and derives conditions on coefficients, including inverse relations and special periodic cases.

Findings

Explicit formulas for solutions in terms of coefficients and parameters

Conditions on coefficients for solution validity and periodicity

Identification of special cases with completely periodic solutions

Abstract

The explicit solution of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODE s with second-degree polynomial right-hand sides, hence featuring 12 a prior arbitrary (time-independent) coefficients. The solution is explicitly provided if the 12 coefficients are expressed by explicitly provided formulas in terms of 10 a prior arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients is also explicitly solved, but it is found to imply as it were, a posterior that the 12 coefficients must then satisfy 4 algebraic constraints, which are explicitly exhibited.Special sub cases are also identified the general solutions of which are completely periodic with a period independent of the initial data, or are characterized by additional restrictions on the coefficients which identify particularly interesting models.