AN IN-DEPTH STUDY ON THE APPLICATION OF THE DIFFERENTIAL TRANSFORM METHOD FOR EFFECTIVELY SOLVING VARIOUS-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Abstract

This study presents a comprehensive analysis of the Differential Transform Method (DTM) as an effective tool for solving ordinary differential equations (ODEs) of various orders. Emphasis is placed on the method’s ability to handle both linear and nonlinear ODEs without the need for common simplification techniques such as linearization, discretization, or perturbation, which often introduce additional complexities or reduce accuracy. By systematically applying DTM to different classes of ODEs, the study highlights its versatility and accuracy in handling initial value problems across a range of complexities with the solution of the first, second, third and fourth-orders ODEs. Comparative analyses with analytical methods demonstrate the superiority of DTM in terms of computational efficiency and solution accuracy. Additionally, graphical representations of both the analytical solutions and the approximate results obtained using DTM were plotted across various orders to showcase the robustness of the DTM. This article also includes detailed examples illustrating the step-by-step application of DTM, providing insights into its potential for broader applications in engineering, physics, and applied mathematics. The increasing complexity of systems modeled by ODEs in scientific and engineering fields necessitates efficient and accurate methods for obtaining reliable solutions, thereby justifying the need to explore alternative approaches like DTM. The findings underscore the relevance of DTM as a powerful method for solving ODEs of various orders, making it a valuable
addition to the toolbox of researchers and practitioners in the field. Conclusively, the results show that DTM is an efficient, accurate, and reliable method.