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

• Equivalence between Abelʼs equation and dissipative nonlinear ODEs explained in detail.
• Factorization method is useful in this equivalence.
• Chielliniʼs integrability condition is used in designing either dissipation or nonlinearity.
• Examples of such Abel-designed nonlinear integrable equations are given.

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 their 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.