Generalized symmetries, first integrals, and exact solutions of chains of differential equations

TL;DR

This paper explores new integrability properties of differential equation chains, providing a method to find all first integrals and exact solutions for equations like Riccati and Abel chains without integration.

Contribution

It introduces a novel approach to determine all first integrals of differential equation chains using generalized symmetries and Jacobi last multipliers, enabling exact solutions.

Findings

All first integrals of the chains are obtained without integration.

Exact general solutions for Riccati and Abel chains are derived.

The method applies to a broad class of differential equations.

Abstract

New integrability properties of a family of sequences of ordinary differential equations, which contains the Riccati and Abel chains as the most simple sequences, are studied. The determination of n generalized symmetries of the nth-order equation in each chain provides, without any kind of integration, n-1 functionally independent first integrals of the equation. A remaining first integral arises by a quadrature by using a Jacobi last multiplier that is expressed in terms of the preceding equation in the corresponding sequence. The complete set of n first integrals is used to obtain the exact general solution of the nth-order equation of each sequence. The results are applied to derive directly the exact general solution of any equation in the Riccati and Abel chains.