Differential Equations

Equations whose unknowns are functions — the native language of every changing system in physics and biology.

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The brief

When Isaac Newton published the Principia Mathematica in 1687, the equations describing planetary motion that he wrote down — F = ma, applied to gravity — were, in modern terms, differential equations: equations whose unknowns are functions, not numbers, and which constrain those functions through their rates of change. Newton did not use the term, and his methods of solution were geometric and intuitive. The eighteenth century industrialized the technique. Euler, the Bernoulli family, Lagrange, Laplace — between them they developed methods of solution, classification, and analysis that turned differential equations into the native language of physics. By 1800, every major branch of natural science had reformulated itself as a system of differential equations.

A differential equation is an equation involving an unknown function and one or more of its derivatives. The unknown is a function, not a number; the equation specifies how the function changes, and a solution is any function satisfying the relationship. Ordinary differential equations (ODEs) involve a function of a single variable: dy/dt = f(y, t) is a first-order ODE constraining how y evolves over time. Partial differential equations (PDEs) involve multiple variables and their partial derivatives: ∂u/∂t = α·∂²u/∂x² is the heat equation, describing how temperature u(x, t) spreads through a rod. Equations are linear if the unknown function and its derivatives appear only as a linear combination, nonlinear otherwise. Initial-value problems specify the function's value at one point in time and ask how it evolves; boundary-value problems specify values at multiple points and ask what function fits. Existence and uniqueness theorems (Picard–Lindelöf, Cauchy–Kovalevskaya) guarantee solutions under reasonable conditions. Methods of solution are a sprawling craft: separation of variables, integrating factors, Laplace transforms, series solutions, special functions, eigenfunction expansions, and — for almost every equation that matters in practice — numerical methods (Runge–Kutta, finite differences, finite elements). The deep reason differential equations are the language of physics is that most physical laws are local: forces, fields, fluxes, and reaction rates act on infinitesimal neighborhoods, and the derivative is exactly the language of infinitesimal neighborhoods. Famous DEs span all of science: Newton's F = ma, Maxwell's equations of electromagnetism, Schrödinger's equation of quantum mechanics, the heat equation, Navier–Stokes (whose existence and smoothness in 3D is one of the seven Millennium Prize Problems), the Black–Scholes equation of option pricing, the Lotka–Volterra predator–prey equations, the SIR model of epidemics.

Why nowNumerical solution of differential equations is one of the largest categories of computational work in science. Climate models are coupled PDE systems run on supercomputers. Pandemic modeling — visible to the entire public during COVID — runs SIR-family ODEs. Financial derivative pricing solves the Black–Scholes PDE billions of times daily. Robotics control and autopilot systems track and stabilize systems described by ODEs. Drug pharmacokinetics models drug concentration over time as ODEs. Computer graphics fluid simulation solves Navier–Stokes approximately for visual effects. The most successful predictive technology humans have ever built — physics — runs almost entirely on differential equations, and the work of finding, solving, and approximating them is most of applied mathematics.

Further readingBoyce and DiPrima's Elementary Differential Equations (11th ed., 2017) is the standard sophomore-level text; Strang's Differential Equations and Linear Algebra is the more conceptually unified alternative. For partial differential equations, Evans's Partial Differential Equations (2010) is the graduate canon. For the dynamical-systems perspective — phase portraits, attractors, chaos — Strogatz's Nonlinear Dynamics and Chaos (2014) is the field's most beloved book. Arnold's Ordinary Differential Equations (1992) is the rigorous Russian-tradition view.