Commutativity and Commutative Pairs of Some Differential Equations

TL;DR

This paper derives explicit differential equations for commutative pairs of second-order linear systems, investigates their properties, and explores a novel cryptological application for signal obscuring in telecommunications.

Contribution

It provides explicit forms of commutative pairs for certain differential equations and introduces a new cryptology application for signal security.

Findings

Certain second-order systems have commutative pairs with explicit solutions.

Commutativity conditions vary among different systems.

Application in cryptology for signal obscuring is demonstrated.

Abstract

In this study, explicit differential equations representing commutative pairs of some well-known second-order linear time-varying systems have been derived. The commutativity of these systems are investigated by considering 30 second-order linear differential equations with variable coefficients. It is shown that the system modeled by each one of these equations has a commutative pair with (or without) some conditions or not. There appear special cases such that both, only one or neither of the original system and its commutative pair has explicit analytic solution. Some benefits of commutativity have already been mentioned in the literature but a new application for in cryptology for obscuring transmitted signals in telecommunication is illustrated in this paper.