Analytical Methods for Solving Mathematical Systems of Ordinary Differential Equations

This study addresses the analytical resolution of systems of ordinary differential equations (ODEs), which are foundational in modeling various dynamic processes across scientific fields. While numerous methods exist, a clear comparative framework for solving linear systems remains underexplored. This paper fills that gap by employing and contrasting three core techniques: the D-operator method, eigenvalue analysis, and integral transforms (especially Laplace). Each method is applied to illustrative examples, demonstrating their efficiency, limitations, and the conditions under which they yield general solutions. The results reveal that integral transforms, particularly Laplace, offer more streamlined solutions for linear systems with initial conditions, while eigenvalue methods excel in homogeneous cases. These findings provide valuable insights for selecting appropriate analytical tools in mathematical modeling and engineering applications.