An In-Depth Study of Cauchy-Euler Differential Equations and Their Numerical Solutions Using MATLAB
DOI:
https://doi.org/10.47134/ppm.v3i1.2083Keywords:
Cauchy-Euler Equation, Differential Equations, Logarithmic Transformation, Homogeneous Equations, Non-Homogeneous Equations, MATLAB, Numerical Methods, Ode45, Characteristic Equation, Analytical SolutionsAbstract
The Cauchy–Euler differential equation, distinguished by its dependence on the independent variable through variable coefficients, represents an essential category of linear differential equations with broad applications across mathematics, physics, and engineering. The present work provides a comprehensive exploration of both homogeneous and non-homogeneous forms of these equations, focusing on their analytical treatments as well as numerical approaches for cases in which explicit closed-form solutions are not easily attainable. Through the use of a logarithmic substitution, reckonings with mutable constants can be distorted into constant-coefficient reckonings, allowing the application of classical strategies such as the characteristic polynomial method and the approach of undetermined coefficients. Illustrative examples are included to show the derivation of both general and particular solutions, covering situations with repeated or complex roots. In addition, the study incorporates numerical techniques—particularly MATLAB’s ode45 solver—to approximate solutions of Cauchy–Euler equations, especially for non-homogeneous problems and systems subject to initial conditions. These implementations highlight the adaptability and efficiency of numerical solvers in scenarios where analytical methods are difficult or infeasible. By combining analytical and computational methodologies, this work provides an integrated framework for addressing a wide range of differential equations encountered in scientific and engineering applications. It also emphasizes the enduring significance of Cauchy–Euler equations and the crucial role of computational platforms such as MATLAB in modern differential equation analysis
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