Applications of Lie systems in dissipative Milne--Pinney equations

TL;DR

This paper applies Lie system theory to dissipative Milne--Pinney equations, establishing a superposition rule that expresses general solutions via particular solutions of linear equations, advancing the analytical methods for such systems.

Contribution

It introduces a geometric approach to dissipative Ermakov systems, deriving a superposition rule for solutions of dissipative Milne--Pinney equations.

Findings

Superposition rule for dissipative Milne--Pinney equations

Expresses solutions in terms of linear differential equations

Advances solution methods for dissipative systems

Abstract

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne--Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants.