On the general solution for the modified Emden-type equation

Abstract

In this paper, we demonstrate that the modified Emden type equation (MEE),ẍ + αxẋ + βx 3 = 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 function 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 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.

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

“…In [10,11] Duarte et al have extended the PS method to include such situations. Subsequently in a series of papers [6,8,9] Chandrasekar et al have uncovered rational and even non rational first integrals for a large class of oscillator type equations, by appropriately modifying and also extending the basic idea behind the PS procedure.…”

“…Most recently Lakshmanan and his coworkers have generalized and used the extended Prelle-Singer method to obtain the first integrals and general solutions for a class of nonlinear equations [6,7,9]. They have also devised a procedure to construct a transformation which removes the time-dependent part from the first integral and provides the general solution by quadrature [8].…”

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

“…The example which we have chosen to illustrate the symmetry methods is the modified Emden equation (MEE) [48,49,50,51,52,53,54,55] …”

“…For example, it arises in the study of equilibrium configurations of a spherical gas cloud acting under the mutual attraction of its molecules and subjected to the laws of thermodynamics [50,51]. Equation (1) admits time independent nonstandard Lagrangian and Hamiltonian structures [53]. In the contemporary literature, this equation has been considered by several authors in different contexts.…”

“…In the contemporary literature, this equation has been considered by several authors in different contexts. For example, Chandrasekar, Senthilvelan and Lakshmanan have studied the linearization and investigated the integrability of this equation through the extended Prelle-Singer procedure [53]. The Lie point symmetries of this equation were also derived by few authors in different contexts.…”