In this paper, we develop an algorithm for solving nonhomogeneous first order linear differential systems by using the Jordan decomposition and convert this algorithm into a Maple procedure to find the exact solution to many-variable systems.



Differential equations play an important role in the understanding of physical sciences. Many differential equations arise from problems in physics, engineering, and other sciences, and these equations serve as mathematical models for solving numerous problems in science and engineering([1]-[11]).Numerous numerical methods exist for solving differential equations, such as Taylor, Picard, Euler, Runge-Kutta and transformation methods.



Using the Jordan decomposition method, we can simplify the exponential matrix into a product of matrices. This allows us to easily find the exact solution to the system.



After the introduction, the paper is divided into six sections. In section 2, we introduce the definition and properties of matrix exponentials. Then, in section 3, we provide an overview of Jordan decomposition for matrices. Section 4 presents our algorithm. In section 5, we obtain the Maple procedure. In section 6 we apply the algorithm to find the exact solution to some linear differential systems. Finally, we concluded our result