Symmetry and Singularity Properties of Steen-Ermakov-Milne-Pinney Equations

TL;DR

This paper investigates the symmetry and singularity characteristics of a class of differential equations related to the Steen-Ermakov-Milne-Pinney equations, revealing algebraic patterns and symmetry structures depending on parameter values.

Contribution

It classifies the Lie point symmetries of a broad class of differential equations and analyzes their singularity properties for different parameter regimes.

Findings

Algebraic structures vary with parameter lpha.

Identifies three main symmetry algebra types.

Provides insight into the singularity behavior of these equations.

Abstract

We examine the general element of the class of ordinary differential equations, $yy^{(n+1)}+\alpha y'y^{(n)}=0$, for its Lie point symmetries. We observe that the algebraic properties of this class of equations display an attractive set of patterns, the general member of the class can have three type of Algebra, $(n+1)A_1 \oplus_s\{A_1 \oplus sl(2,R)\}$, $A_1 \oplus sl(2,R)$ or $A_2 \oplus A_1$, for different values of $\alpha$. We look at the singularity properties of these equations for various values of $\alpha$.