Abstract
The modified Emden-type is being investigated by mathematicians as well as physicists for about a century. However, there exist no exact explicit solution of this equation, ẍ + αxẋ + βx3 = 0 for arbitrary values of α and β. In this work, the exact analytical explicit solution of modified Emden-type (MEE) equation is derived for arbitrary values of α and β. The Lagrangian and Hamiltonian of MEE are also worked out. The solution is also utilized to find exact explicit analytical solution of Force-free Duffing oscillator-type equation. And exact explicit analytical solution of two-dimensional Lotka-Volterra System is also worked out.
Highlights
The modified Emden-type equation [MEE] or the modified Painleve-Ince equation is x + α xx + β x3 = 0 (1.1)where the dot represents derivative with respect to time and α, β are arbitrary parameters
Himself obtained solutions of (1.1) for α2 = 9β and α2 = –β. This equation is extremely important because it arises in many mathematical problems like univalent functions defined by differential equation of second order [4]
The MEE is a special case of one dimensional analogue [6, 7] of gauze boson theory introduced by Yang and Mills
Summary
Himself obtained solutions of (1.1) for α2 = 9β and α2 = –β This equation is extremely important because it arises in many mathematical problems like univalent functions defined by differential equation of second order [4]. As regards solution of MEE i.e., Equation (1.1), some progress in recent past has been done by Chandrasekar et al [10]. These authors have made good contributions by finding Lagrangian, Hamiltonian and Invariant of (1.1) for distinct cases: 1) α2 = 8β, 2) α2 > 8β and 3) α2 < 8β. Equation (2.23) gives the exact analytic explicit solution of Emden-type Equation (1.1) for arbitrary values of α and β. In (2.23) A and B are two arbitrary constants
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