Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

TL;DR

This paper explores connections between dissipative nonlinear second order differential equations and Abel equations, using factorization and Chiellini's criterion to identify integrable cases with applications to biological and physical models.

Contribution

It introduces a novel approach to find integrable dissipative equations through factorization and Abel equations, expanding the toolkit for solving nonlinear differential equations.

Findings

Identifies new integrable dissipative equations using Abel and factorization methods.

Provides explicit solutions for Fisher, nonlinear pendulum, and Burgers-Huxley equations.

Demonstrates how to derive Abel solutions directly from second-order equation factorizations.

Abstract

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second-order nonlinear equations