Differential equation

A differential equation is a mathematical equation that relates a function with its derivatives, encoding how a quantity changes in response to itself and its environment. They are the native language of physics and engineering: Newton's second law is a differential equation relating force to acceleration; Maxwell's equations are differential equations relating electric and magnetic fields; the spread of disease is modeled by differential equations coupling infection rates to population states.

The classification of differential equations shapes the methods available to solve them. Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives; partial differential equations (PDEs) involve functions of multiple variables and partial derivatives. Linear ODEs with constant coefficients yield to the Laplace transform, converting differential operators into algebraic multipliers. Nonlinear equations generally resist closed-form solution and demand numerical methods or qualitative analysis through dynamical systems theory.

The differential equation is the original sin of mathematical modeling: it presumes that the future is determined by the present and its rate of change, a assumption of infinite differentiability and local causality that breaks down at quantum scales, across shock waves, and in any system where history matters in ways that cannot be compressed into a derivative.