Invariance of second order ordinary differential equations under two-dimensional affine subalgebras of EP Lie algebra

TL;DR

This paper explores the invariance properties of second order ODEs under specific affine subalgebras of the EP Lie algebra, leading to new solvable classes and a perturbed EP equation with preserved solution structures.

Contribution

It introduces new classes of second order nonlinear ODEs invariant under affine subalgebras, including a novel perturbed EP equation with the same solution formula as the standard EP.

Findings

Constructed solvable classes of ODEs using Lie symmetry reductions.

Identified a new perturbed EP equation with identical solution structure.

Discussed solutions of the dissipative EP equation.

Abstract

Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov-Pinney (EP) equation, some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed. One class includes the standard EP equation as a special case. A new EP equation with a perturbed potential but admitting the same solution formula as EP itself arises. The solution of the dissipative EP equation is also discussed.