Using the generalized sundman transformation to achieve linearization of the general modified second-order lane-emden differential equation

Abstract

The generalized Sundman transformation is a mathematical technique designed to simplify the integration of differential equations, particularly in fields like dynamical systems and celestial mechanics. This powerful method helps transform complicated dynamical equations into forms that are easier to analyze or solve numerically, especially when dealing with challenging singularities. Among the various nonlinear second-order differential equations, the general modified Emden equation (GMEE) is notable for its frequent appearance across multiple areas of applied mathematics and physics. This equation is a variation of the classic Emden-Fowler equation, which is commonly used to model thermodynamics, stellar structure, and other physical phenomena. Its nonlinear nature allows it to effectively represent the complexities found in real-world systems across diverse fields, making it highly versatile. This study examines the generalized modified Lane-Emden equation derived from the general Lane-Emden differential equation. Using the generalized Sundman transformation approach, exact solutions are obtained for the second-order general modified Lane-Emden differential equation through analytical linearization. Additionally taken into consideration were a few particular instances of the modified Lane-Emden differential equations and their solutions.