Solving Linear Ordinary Differential Equations with Constant Coefficients via Inverse Laplace Transform and Residue Theorem

Abstract

This research paper discusses the advanced mathematical framework of the Residue Theorem within complex analysis, combined with the Laplace Transform, as a precise analytical methodology for solving linear ordinary differential equations with constant coefficients. The research aims to address the challenges associated with finding the inverse Laplace transform for complex systems with high-order or multivalued poles in the complex plane. The methodology relies on transferring the differential equation from the time domain t to the complex frequency domain s to convert it into a purely algebraic equation, and then utilizing the Bromwich Integral and the Residue Theorem to accurately compute the inverse Laplace transform and achieve closed-form solutions. The study demonstrates that integrating the Residue Theorem with the Laplace Transform not only simplifies mathematical solutions but also provides a crucial tool for understanding the stability and dynamic behavior of engineering and physical systems under various initial conditions.