Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides

TL;DR

This paper identifies a specific class of two-dimensional nonlinear ODE systems with homogeneous quadratic terms that are explicitly solvable, characterized by algebraic constraints on parameters, and extends these results to nonhomogeneous cases.

Contribution

It provides explicit algebraic conditions for solvability of a special class of quadratic two-dimensional systems and extends the analysis to nonhomogeneous and isochronous cases.

Findings

Explicit algebraic constraints for solvability

Extension to nonhomogeneous quadratic systems

Identification of isochronous cases

Abstract

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases.