Equation of Change : First-Order Linear Ordinary Differential Equations in Mechanical and Thermal System

Abstract : This paper presents a focused study on the role of first-order linear ordinary differentialequations (ODEs) as equations of change in both mechanical and thermal systems. Emphasizing theirfoundational significance, the paper explores how these equations model time-dependent processes inclassical mechanics and thermodynamics. In mechanical systems, applications include falling objectswith air resistance, damped oscillations, and simple electrical circuits, where first-order ODEs describethe evolution of velocity, displacement, and current under resistive forces. In thermal systems, thestudy examines Newton’s law of cooling, temperature variations in interconnected bodies, and thebehavior of fluids under thermal influence, highlighting the predictive power of first-order models indescribing heat flow and thermal equilibrium. Each scenario demonstrates how physical laws—such asNewton’s laws of motion or Fourier’s law of heat conduction—translate naturally into first-orderdifferential equations. Through analytical solutions and physical interpretation, this work illustrateshow equations of change unify diverse phenomena under a common mathematical framework,reinforcing their importance in modeling, analysis, and engineering design.