Payame Noor University (PNU) Control and Optimization in Applied Mathematics 2383-3130 10 2 2025 07 01 A Computer Algebra Approach to Linear ODE Systems with Parametric Coefficients 135 152 12225 10.30473/coam.2025.74498.1307 EN Mahdi Dehghani Darmian Department of Basic Sciences‎, ‎Technical and Vocational University (TVU)‎, ‎Tehran‎, ‎Iran‎. Journal Article 2025 02 07 This paper analyzes systems of linear first-order ordinary differential equations (ODEs) with parametric coefficients‎, ‎a class of problems that arises in control theory‎, ‎optimization‎, ‎and applied mathematics‎. ‎We introduce the notion of a comprehensive solution system for such parametric ODEs, constructed using Gröbner systems from computer algebra‎. ‎ Our approach partitions the parameter space into finitely many cells and associates an explicit solution with each cell‎. ‎ Furthermore, ‎we present an algorithm that computes a comprehensive solution system for any given parametric system‎. ‎To address the computational challenges inherent in Gröbner systems‎, ‎we adopt the GES algorithm‎, ‎a parametric variant of Gaussian elimination‎, ‎which eliminates the need for Gröbner bases. This method builds upon the LDS algorithm proposed in 2017‎. ‎ Both algorithms have been implemented in Maple‎, ‎and we illustrate the structural framework of the main algorithm with a straightforward example. The results highlight the practicality and effectiveness of the proposed methods for solving parametric linear first-order ODE systems. https://mathco.journals.pnu.ac.ir/article_12225_6500e1c59f0957911007c2d962efc509.pdf