An Integrated Transform-Algebraic Framework for Solving Linear Nonautonomous Dynamical Systems

Abstract

This investigation develops a comprehensive algebraic methodology for treating linear nonautonomous ordinary differential equation systems through the synergistic combination of integral transform techniques and matrix elimination procedures. The proposed framework converts the temporal initial value problem expressed as  with  into an equivalent algebraic formulation within the complex frequency domain characterized by . The time-domain solution emerges following the application of the inverse integral transform. This strategy circumvents explicit integration procedures, captures the system dynamics via the resolvent operator , and furnishes a systematic computational protocol that seamlessly accommodates prescribed initial states. The exposition details the theoretical foundations, illustrates the operational sequence through elaborated case studies, and critically examines both the strengths and constraints of the technique, thereby confirming its utility across applied mathematics, engineering disciplines, and allied scientific domains.