On solutions of singular differential equations of the second order

TL;DR

This paper investigates the behavior of solutions to second-order ordinary differential equations with singular points where coefficients vanish, focusing on solutions entering singular points without a definite tangential direction, especially for cubic polynomial right-hand sides.

Contribution

It provides new insights into the behavior of solutions near singular points for a class of second-order differential equations with polynomial nonlinearities.

Findings

Characterization of solution behavior near singular points

Analysis of solutions with cubic polynomial right-hand sides

Insights into solutions entering singular points without fixed tangential directions

Abstract

We study the behaviour of solutions of ordinary differential equations of the second order with singular points, where the coefficients of the second-order derivative vanishes. In particular, we consider solutions entering a singular point without definite tangential direction. Great attention is paid to second-order equations, whose right-hand sides is a cubic polynomial by the first-order derivative.