Exact solutions of the generalized Lane–Emden equation

Abstract

A generalized Lane-Emden equation with indices (␣,␤,,n) is discussed, which reduces to the Lane-Emden equation proper for ␣ϭ2, ␤ϭ1, ϭ1. General properties of the set of solutions of this equation are derived, and exact solutions are given. These include a singular solution without free integration constant for arbitrary n and for particular relations between n, , and ␣. Among the two-parameter solutions nonequivalent families of solutions of the same equation are obtained.

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

“…where α, β, ν and n are real, has been recently looked at in Goenner and Havas [18] and Goenner [19]. In Goenner [19], the author uncovered symmetries of eqn (1.6) to explain integrability of (1.6) for certain values of the parameters considered in Goenner and Havas [18].…”

“…In Goenner [19], the author uncovered symmetries of eqn (1.6) to explain integrability of (1.6) for certain values of the parameters considered in Goenner and Havas [18]. The reader is also referred to the works ( [23], [24], [6], [7]) for symmetries and solutions of Emden-type equations.…”

Simple models for the distribution of dark matter

“…When the sum contains only a single term, this is a particular case of a family of differential equations investigated by Goenner & Havas (2000), which they called the generalized Lane-Emden equations. In fact, it corresponds to the special case of parameter combinations noted by them (see Eq.…”

Analytical solutions to a quasilinear differential equation related to the Lane–Emden equation of the second kind

“…Recently, the Lane-Emden equation of the second kind has generated interest; see Momoniat and Harley (2006), Harley and Momoniat (2008), Goenner and Havas (2000) and the references therein. For a quick derivation of the Lane-Emden equation of the second kind, consider the hydrostatic equation gravitational force on the rhs balanced by the pressure gradient on the lhs d P dr = − M(r )ρ(r )G r 2 , (1.2) and the definition of density ρ in spherical geometry given by…”