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

摘要

In this paper, we consider a second order nonlinear ordinary differential equation of the form xuml+k(1)(x center dot(2)/x)+(k(2)+k(3)x)x center dot+k(4)x(3)+k(5)x(2)+k(6)x=0, where k(i)'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) k(2)=0, k(3)=0 and (ii) k(1)=0, k(2)=0, k(5)=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.