Two integrable classes of Emden-Fowler equations with applications in astrophysics and cosmology

Abstract

We show that some Emden-Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d^2z/d{\chi}^2=A \chi^{-\lambda-2}z^n for \lambda=(n-1)/2 (class one), and \lambda=n+1 (class two). We find their corresponding invariants which reduce them to first order nonlinear ordinary differential equations. Using particular solutions of such EF equations, the two classes are set in the autonomous nonlinear oscillator form d^2\nu/dt^2 +ad \nu/dt +b(\nu-\nu^n)=0, where the coefficients a,b depend only on \lambda, n. For both classes, we write closed-form solutions in parametric form. The illustrative examples from astrophysics and general relativity correspond to two n=2 cases from class one and two, and one n=5 case from class one, all of them yielding Weierstrass elliptic solutions. It is also noticed that when n=2, EF equations can be studied using the Painlev\'e reduction method, since they are a particular case of equations of the type d^2z/d{\chi}^2=F(\chi)z^2, where F(\chi) is the Kustaanheimo-Qvist function