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The resolution of systems of first-order ordinary differential equations (ODEs) stands as a pivotal pursuit with extensive implications across scientific and engineering domains. In tackling this fundamental task, this study undertakes a rigorous comparative assessment of two semi-analytic methodologies, the Variational Iterative Method (VIM) and the New Iterative Method (NIM). Motivated by the need to address a critical research gap, our investigation delves into these approaches' relative merits and demerits. Firstly, it conducts a comprehensive examination of VIM, a well-established method, juxtaposed with NIM, a relatively unexplored approach, to uncover their comparative strengths and limitations. Secondly, the study contributes to the existing knowledge in numerical methods for ODEs by shedding light on essential performance characteristics, including convergence properties, computational efficiency, and accuracy, across a diverse array of ODE systems. Through meticulous numerical experimentation, we not only reveal practical insights into the efficacy of VIM and NIM but also bridge a significant knowledge gap in the field of numerical ODE solvers. Our findings highlight VIM as the more effective method, thus advancing our understanding of semi-analytic approaches for solving ODE systems and furnishing valuable guidance for practitioners and researchers in selecting the most suitable method for their specific applications
Audu et al. (Sun,) studied this question.