An analytical-numerical approach to the Emden-Fowler equation

Abstract

We investigate the Emden-Fowler equation $u^{\prime \prime }=-x^r{u}^p,u(0)=1,u^{\prime }(0)=0$, with roots in astrophysics, and study the qualitative and quantitative dependence of its solution on the parameters; the analytical work is paralleled by numerical simulations. A special attention is given to estimating the first zero $x_0$ of $u$ in terms of $r,p$. The results are used to address two apparently new issues: first, we solve EF backwards starting from $x_0$; second, we transform it into an overdetermined boundary value problem and decide when is this solvable.