A criterion for oscillations in the solutions of the polytropic Lane-Emden equations

We have previously formulated a simple criterion for deducing the intervals of oscillations in the solutions of second-order linear homogeneous differential equations. In this work, we extend analytically the same criterion to superlinear Lane-Emden equations with integer polytropic indices n>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n > 1$\end{document}. We confirm the validity of the analytical results by solving numerically both the cylindrical and the spherical Lane-Emden equations subject to the usual astrophysical boundary conditions for self-gravitating fluids.