A Study of Mathematical Modeling Using Differential Equations in Real-World Systems

Abstract

Differential equations (DEs) constitute one of the most powerful mathematical frameworks for describing, analyzing, and predicting real-world dynamic phenomena. This paper presents a comprehensive investigation of how ordinary differential equations (ODEs) and systems of ODEs are applied across five distinct domains: population dynamics, infectious disease modeling, thermodynamic cooling, electrical circuit behavior, and predator-prey ecological systems. For each application, we derive the governing differential equation, establish initial and boundary conditions, obtain analytical or semi-analytical solutions, and validate results against empirical or synthetic observational data. Detailed numerical results are compiled in five structured comparison tables. Our findings demonstrate that first-order linear and nonlinear ODEs achieve prediction accuracies ranging from 96.6% to 99.9%, with mean absolute percentage errors (MAPE) below 3.5% across all studied applications. The logistic population model outperforms the Malthusian exponential model, the SIR epidemiological system correctly predicts epidemic peak timing within ±2 days, and Newton's cooling model fits experimental data with a root-mean-square error (RMSE) of 0.31°C. These results affirm the indispensability of differential equations as a modeling paradigm in both pure and applied sciences.