Concept
Equations relating functions to their derivatives — the language of laws of change.
Understand it in one breath
Write relations among rates of change as equations. Differential equations model population growth, radioactive decay, planetary motion, and epidemic spread. Newtonian mechanics, Schrödinger's equation, Maxwell's equations, and Navier–Stokes are prominent examples, while some phenomena are better described by stochastic, discrete, or data-driven models.
At a glance
Key formula
Key moments
Principia — mechanics becomes differential law
Newton’s laws of motion and universal gravitation made changing quantities and their rates the language of mechanics, opening the age of differential equations in physics.
d’Alembert’s wave equation
d’Alembert derived a partial differential equation for a vibrating string, beginning the mathematical description of waves in sound and light.
Maxwell’s equations
Maxwell’s theory unified electricity, magnetism, and optics in a system of differential equations and predicted that light is an electromagnetic wave.
Black–Scholes — differential equations reach Wall Street
The Black–Scholes model used a differential equation to describe option prices, helping launch modern quantitative finance.
Modern applications
Newtonian mechanics, Maxwell’s equations, the Schrödinger equation, Navier–Stokes fluids, Black–Scholes option pricing, and SIR epidemic models — the basic language of models across nature and society.
Beyond MathVoyage
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