@article{article_779727, title={Commutativity Conditions of Heine’s Differential Equation}, journal={Avrupa Bilim ve Teknoloji Dergisi}, pages={239–243}, year={2020}, DOI={10.31590/ejosat.779727}, url={https://izlik.org/JA33ZE26LF}, author={Köksal, Mehmet Emir}, keywords={Komütativite, Analog Sistemler, Heine Diferansiyel Denklemi, Zamanla Değişen Sistemler}, abstract={The realization of many engineering systems consists of cascade connection of systems of simple orders, which is very important in design of electrical and electronic systems. Although the order of connection of the systems mainly depends on the special design approach, engineering ingenuity, traditional synthetic methods, when the sensitivity, stability, linearity, noise disturbance, robustness effects are considered the change of the order of connection without changing the main function of the total system (commutativity) may lead positive results. Therefore, the commutativity is very important from the practical point of view. In this paper, commutativity of Heine differential equation is considered. It is shown that the system modeled by a Heine differential equation has a commutative pair or not depending on the parameters of the equation. The conditions that must be satisfied for the commutativity are set in this contribution. An example is considered for application. The initial conditions are assumed to be zero in this study. For the nonzero initial conditions (or for unrelaxed systems), some further relations should hold in addition to the commutativity conditions for relaxed Heine differential systems derived in this paper. The case of nonzero initial conditions do not gain original results on the paper since they have already been studied in the literature (Koksal, 2019) for general second-order continuous-time linear time-varying systems. The commutative pairs of Heine differential equation are obtained for some parameters satisfying the existence conditions of commutativity and it is seen that they are not of Heine type. So, the commutative pairs of Heine differential systems are not Heine type for the general choice of arbitrary constants relating the commutative pairs.}, organization={The Scientific and Technological Research Council of Turkey}