The general forms of the Emden, modified Emden and Emden/Fowler ODEs are given by the following:

Emden_ode := diff(x^2*diff(y(x),x),x)+x^2*y(x)^n=0;

modified_Emden_ode := diff(diff(y(x),x),x)+a(x)*diff(y(x),x)+y(x)^n = 0;

Emden_Fowler_ode := diff(x^p*diff(y(x),x),x)+x^sigma*y(x)^n=0;

where n is an integer and a(x) is an arbitrary function of x.

See Leach, "First Integrals for the modified Emden equation"; and Rosenau, "A Note on Integration of the Emden-Fowler Equation". There are certain special cases of the Emden-Fowler equation which can be solved exactly. See also Polyanin and Zaitsev, "Exact Solutions of Ordinary Differential Equations", p. 241.