COMPARATIVE ANALYSIS OF THE LAPLACE TRANSFORM AND THE FOURIER TRANSFORM FOR SOLVING SECOND-ORDER NON-HOMOGENEOUS DIFFERENTIAL EQUATIONS

Abstract

This paper conducts a comparative analysis of two principal integral transform methods, the Laplace transform and the Fourier transform, for deriving analytical solutions to second-order linear non-homogeneous ordinary differential equations (ODEs). The study systematically examines their theoretical underpinnings, procedural methodologies, respective domains of applicability, and computational efficacy. Although both techniques reduce differential equations to more tractable algebraic forms, they exhibit fundamental distinctions in their operational domains, treatment of initial conditions, and suitability for various forcing functions.

The Laplace transform proves to be the more versatile and powerful tool for solving initial value problems (IVPs), which are prevalent in dynamical systems analysis. Its strength lies in its inherent capacity to incorporate initial conditions at   directly into the transformed equation and its robustness when dealing with discontinuous or exponential-type forcing functions.

Conversely, the Fourier transform is predominantly suited for problems defined on the entire real line . It emerges as the preferred method for steady-state analysis and for handling problems involving periodic or non-decaying signals, particularly those situated on infinite domains.