A Comparative Analysis of the Laplace Transform and Homotopy Perturbation Method for Fourth Order Ordinary Differential Equations

Fourth order ordinary differential equations arise naturally in many branches of science and engineering, including beam deflection theory, vibration analysis, elastic stability, and control engineering. Owing to the presence of higher order derivatives, such equations are often challenging to solve using conventional analytical techniques. This paper presents a clear and fully original comparative analysis of two widely used solution approaches: the Laplace Transform method and the Homotopy Perturbation Method (HPM). The Laplace Transform converts a differential equation into an algebraic form, allowing initial conditions to be incorporated directly and yielding exact closed form solutions for linear problems. In contrast, HPM provides approximate analytical solutions expressed as rapidly convergent series and is particularly effective for nonlinear equations. Several fourth order ordinary differential equations are examined using both methods, and the results are compared in terms of simplicity, computational effort, and accuracy. The study demonstrates that the Laplace Transform is more efficient and straightforward for linear fourth order equations, whereas HPM offers greater flexibility for nonlinear models. This work is intended to guide students and researchers in selecting an appropriate technique for solving higher order differential equations.