On the general solution for the modified Emden type equation $\ddot{x}+\alpha x\dot{x}+\beta x^3=0$

Abstract

In this paper, we demonstrate that the modified Emden type equation (MEE), $\ddot{x}+\alpha x\dot{x}+\beta x^3=0$, is integrable either explicitly or by quadrature for any value of $\alpha$ and $\beta$. We also prove that the MEE possesses appropriate time-independent Hamiltonian function for the full range of parameters $\alpha$ and $\beta$. In addition, we show that the MEE is intimately connected with two well known nonlinear models, namely the force-free Duffing type oscillator equation and the two dimensional Lotka-Volterra (LV) equation and thus the complete integrability of the latter two models can also be understood in terms of the MEE.