Generalized symmetries, first integrals, and exact solutions of chains 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.
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Cite
@article{arxiv.2104.04800,
title = {Generalized symmetries, first integrals, and exact solutions of chains of differential equations},
author = {C. Muriel and M. C. Nucci},
journal= {arXiv preprint arXiv:2104.04800},
year = {2023}
}
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16 pages