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

Resumen

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 ofngeneralized symmetries of thenth-order equation in each chainprovides, without any kind of integration,n−1 functionally independent first integralsof the equation. A remaining first integral arises by a quadrature by using a Jacobi lastmultiplier that is expressed in terms of the preceding equation in the correspondingsequence. The complete set ofnfirst integrals is used to obtain the exact generalsolution of thenth-order equation of each sequence. The results are applied to derivedirectly the exact general solution of any equation in the Riccati and Abel chains