Analytic and numerical solutions to the Lane–Emden equation

In this Letter, we present analytical solutions to the Lane–Emden equation y″(x)+2xy′(x)+f(y)=0 describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solutions are obtained by using the traditional power series approach and by using the Homotopy Analysis Method (HAM). We show that the series solutions obtained by the Homotopy Analysis Method converge in a larger interval than in the case of the corresponding traditional series solutions. Furthermore, we obtained numerical solutions (using Runge–Kutta–Fehlberg 4-5 technique) which are used to validate the analytical solutions.