A Hybrid Analytical-Geometric Methodology for Autonomous Linear Dynamical Systems

Abstract

This investigation advances an integrated analytical and qualitative examination of autonomous linear differential systems, which constitute a fundamental paradigm for modeling dynamical phenomena across engineering, physics, and applied mathematics. The primary goal is to establish a comprehensive characterization of solution behavior and stability through a unified mathematical framework. The proposed methodology synergistically combines the Laplace transform with the Gauss-Jordan elimination technique to derive explicit closed-form solutions for coupled linear systems. This hybrid analytical protocol not only simplifies the solution procedure but also improves computational efficiency when addressing systems of elevated complexity. Furthermore, the qualitative evolution of solutions undergoes rigorous scrutiny via eigenvalue and eigenvector analysis. The findings demonstrate that the spectral characteristics of the system matrix play a decisive role in determining stability properties and asymptotic system dynamics. Equilibrium configurations receive systematic classification into stable nodes, unstable nodes, and saddle points, substantiated by geometric interpretation through phase plane analysis. A comparative evaluation of solution methodologies-encompassing the Laplace transform technique, eigenvalue-based algebraic approaches, and computational approximation methods-is additionally presented. The outcomes highlight that while analytical approaches furnish exact solutions and profound theoretical insight, numerical techniques afford flexibility in managing complex and large-scale systems. The integration of these distinct paradigms yields a more robust and holistic framework for dynamical system analysis.