Explicitly Solvable Systems of First-order Ordinary Differential Equations with Polynomial Right-hand Sides, and Their Periodic Variants

TL;DR

This paper identifies special classes of first-order ODE systems with polynomial right-hand sides that have explicit solutions or periodic solutions, based on constraints involving parameters and initial data.

Contribution

It introduces a novel class of explicitly solvable and periodic first-order ODE systems with polynomial right-hand sides, considering parameter and initial data constraints.

Findings

Explicit solutions for systems with homogeneous polynomial right-hand sides

Periodic solutions for non-homogeneous variants

Characterization of systems via parameter and initial data constraints

Abstract

In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data.