Application of Ordinary Differential Equations in Real-World Problems

Ordinary differential equations (ODEs) are among the most powerful mathematical tools for modelling physical, biological, chemical, and engineering phenomena. This paper presents a systematic study of the formulation, analytical solution, and real-world application of first-order and second-order ordinary differential equations. Three major application areas are investigated in detail: (i) exponential growth and radioactive decay governed by the first-order linear ODE dy/dt = ky, (ii) Newton's Law of Cooling modelled by the first-order ODE dT/dt = -k(T - T_a), and (iii) simple harmonic motion described by the second-order ODE d²x/dt² + ω²x = 0. For each application, the differential equation is derived from physical principles, solved analytically using separation of variables and integrating factor methods, and the solution is verified numerically with tabulated results. Worked examples with data are presented for population growth, carbon-14 radioactive dating, cooling of a heated metal object, and spring-mass vibration. The paper demonstrates that ODEs provide accurate quantitative predictions for diverse real-world problems and serves as a comprehensive introductory reference for polytechnic and undergraduate students of applied mathematics.