Two Integrable Classes of Emden–Fowler Equations with Applications in Astrophysics and Cosmology

Abstract

Abstract We show that some Emden–Fowler (EF) equations encountered in astrophysics and cosmology belong to two EF integrable classes of the type d 2 z / d χ 2 = A χ − λ − 2 z n ${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=A{\chi^{-\lambda-2}}{z^{n}}$ for λ = ( n − 1 ) / 2 $\lambda=(n-1)/2$ (class 1), and λ = n + 1 $\lambda=n+1$ (class 2). 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 the form d 2 ν / d t 2 + a d ν / d t + b ( ν − ν n ) = 0 ${\mathrm{d}^{2}}\nu/\mathrm{d}{t^{2}}+a\mathrm{d}\nu/\mathrm{d}t+b(\nu-{\nu^{n}})=0$ , where the coefficients a , b $a,b$ depend only on λ , n $\lambda,n$ . 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 1 and 2, and one n = 5 case from class 1, all of them yielding Weierstrass elliptic solutions. It is also noticed that when n = 2, the EF equations can be studied using the Painlevé reduction method, since they are a particular case of equations of the type d 2 z / d χ 2 = F ( χ ) z 2 ${\mathrm{d}^{2}}z/\mathrm{d}{\chi^{2}}=F(\chi){z^{2}}$ , where F ( χ ) $F(\chi)$ is the Kustaanheimo-Qvist function.