Exact solutions of the generalized Lane–Emden equation

A generalized Lane–Emden equation with indices (α,β,ν,n) is discussed, which reduces to the Lane–Emden equation proper for α=2, β=1, ν=1. 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, ν, and α. Among the two-parameter solutions nonequivalent families of solutions of the same equation are obtained.