Commutativity and Commutative Pairs of Some Differential Equations
For researchers in differential equations and cryptology, this work provides a catalog of commutative pairs and a novel application, though the results are incremental and domain-specific.
The study derives commutative pairs for 30 second-order linear time-varying differential equations, identifying conditions under which commutativity holds, and demonstrates a new application in cryptology for obscuring telecommunication signals.
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.