On certain new integrable second order nonlinear differential equations and their connection with two dimensional Lotka-Volterra system

Abstract

In this paper, we consider a second order nonlinear ordinary differential equation of the form x+k₁(x²/x)+(k₂+k₃x)x+k₄x³+k₅x²+k₆x = 0, where ki's, i = 1,2,...,6, are arbitrary parameters. By using the modified Prelle-Singer procedure, we identify five new integrable cases in this equation besides two known integrable cases, namely (i) k₂ = 0, k₃ = 0 and (ii) k₁ = 0, k₂ = 0, k₅ = 0. Among these five, four equations admit time-dependent first integrals and the remaining one admits time-independent first integral. From the time-independent first integral, nonstandard Hamiltonian structure is deduced, thereby proving the Liouville sense of integrability. In the case of time-dependent integrals, we either explicitly integrate the system or transform to a time-independent case and deduce the underlying Hamiltonian structure. We also demonstrate that the above second order ordinary differential equation is intimately related to the two dimensional Lotka-Volterra system. From the integrable parametric choices of the above nonlinear equation all the known integrable cases of the LV system can be deduced.