A variant of frobeniu' method for the emden fowler equation

Abstract

In this note, special solutions of the Emden Fowler equations are constructed by Puiseux expansions. The approach is a generalisation of the Frobenius theory at regular singular points of linear analytic differential equations. Convergence of the expansions is proved by a contraction argument. It is shown that two different families of solutions are constructible via Puiseux expansions. The solutions reduce to Bessel functions for exponent β = 1, where the equation gets linear, and to the homoclinic loop at the second critical exponent β2. The relation to the seperatrices from the phase plane systems introduced by Rothe in [17], as well as van den Broek and Verhulst in [5] is established. One of the two families of solutions are just the well known (E) solutions arising in the description of selfgravitating polytropic gasspheres. The mass radius relation of polytropic gasspheres as discussed in Shapiro and Teukolsky [18] is extended to exponents from zero up to the second critical exponent.