On the complete integrability and linearization of nonlinear ordinary differential equations - Part III: Coupled first order equations

Continuing our study on the complete integrability of nonlinear ordinary differential equations, in this paper we consider the integrability of a system of coupled first order nonlinear ordinary differential equations (ODEs) of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle-Singer procedure so as to apply it to both autonomous and non-autonomous systems of coupled first order ODEs. We briefly explain the method of finding integrals of motion (time independent as well as time dependent integrals) for two and three coupled first order ODEs by extending the Prelle-Singer(PS) method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two dimensional Lotka-Volterra system and three-dimensional R$\ddot{o}$ssler system as well as other examples including non-autonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first order ODEs.