On the general solution for the modified Emden-type equation x^.. + αxx^. + βx³ = O

Abstract

In this paper, we demonstrate that the modified equation of Emden type (MEE), x^.. + αxx^. + βx³ = 0, is integrable either explicitly or by quadrature for any value of α and β . We also prove that the MEE possesses appropriate time-independent Hamiltonian functions for the full range of parameters α and β. In addition we show that the MEE is intimately connected with two well-known nonlinear models, namely the equation of force-free Duffing-type oscillator and the two-dimensional Lotka-Volterra equation, and thus the complete integrability of the latter two models can also be understood in terms of the MEE.