Research Article AGeneralSchemeforSolvingSystemsofLinearFirst-Order DifferentialEquationsBasedontheDifferential Transform Method AhmedHusseinMsmali , 1 A.M.Alotaibi , 2 M.A.El-Moneam , 1 BadrS.Badr, 1 andAbdullahAliH.Ahmadini 1 1 Mathematics Department, Faculty of Science, Jazan University, Jazan, Saudi Arabia 2 Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia Correspondence should be addressed to M. A. El-Moneam; mabdelmeneam2014@yahoo.com Received 25 August 2020; Accepted 17 August 2021; Published 27 August 2021 Academic Editor: Efthymios G. Tsionas Copyright © 2021 Ahmed Hussein Msmali et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. e differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. en, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method. 1.Introduction e differential transform was first introduced by Zhou [1], and up to now, the DT method has been developed for solving various kinds of differential and integral equations in many literatures. For example, Ali [2] has developed the DT method for solving partial differential equations and Ayaz [3, 4] has applied this method to differential algebraic equations. Arikoglu and Ozkol [5] have solved the inte- grodifferential equations with boundary value conditions by the DT method. Odibat [6] has used the DT method for solving Volterra integral equations with separable kernels. Tari and Ziyaee [7] have solved the system of two-dimen- sional nonlinear Volterra integrodifferential equations by the DT method. e systems of integral and integrodiffer- ential equations, the multiorder fractional differential equations, the system of fractional differential equations, the singularly perturbed Volterra integral equations, and the time-fractional diffusion equation have been solved by the DT method in [2, 6, 8–10]. Also, the DT method has been applied to nonlinear parabolic-hyperbolic partial differential equations, and a modified approach of DT has been de- veloped to nonlinear partial differential equations in studies by Biazarand Abdul Halim-Haasan [8]. Patil and Kembayat [11] have solved the two-dimensional Fredholm integral equations. Abdewahid [12] introduced a new basic formula s for the one-dimensional differential transform. e main aim of this work is to introduce new useful algorithms depending on the DT method to solve systems of linear differential equations. 2.AnalysisofDifferentialTransform e basic definition and the fundamental theorems of the DTM and its applicability for various kinds of differential equationsaregivenin[13–20].Fortheconvenienceofthe reader, we will present a review of the DTM. To do this, we assume that f(x) ∈ C ∞ (I); then, for any point Hindawi Journal of Mathematics Volume 2021, Article ID 8839201, 9 pages https://doi.org/10.1155/2021/8839201