Differential equations

Ordinary algebra solves for an unknown number. A differential equation solves for an unknown *function*, given a relation that ties the function to its own rates of change. "The rate a quantity grows is proportional to how much there is" is the differential equation dy/dt = k·y, and its solution is the exponential y = e^{kt}. Cooling bodies, hanging chains, vibrating strings, orbiting planets — each is captured by such a relation. Equations in one variable are ordinary (ODEs); those in several, like heat spreading through a solid, are partial (PDEs). The subject crystallized around 1690: solving the isochrone in the *Acta Eruditorum*, Jacob Bernoulli wrote down the governing first-order equation and cracked it by separation of variables — and in that very paper the word "integral" first appears in print with its modern meaning. Leibniz gave the new objects their name, *aequatio differentialis*.

A law of nature almost never tells you where something *is*; it tells you how it *changes*. A differential equation is the exact container for that kind of statement, and solving it reconstructs the full behavior from the rule of change. That is why so much of quantitative science is, literally, a list of differential equations: Newtonian mechanics, the conduction of heat, the propagation of waves and light, Maxwell's electromagnetism, the flow of fluids, chemical kinetics, population dynamics, the pricing of financial options. To possess the theory of differential equations is to possess the grammar in which physical law is written.

Differential equations are the working machinery of physics and every branch of engineering. Downstream they grew into the calculus of variations, dynamical systems and chaos theory (where tiny differences in starting point explode into wholly different futures), control theory, and the vast numerical methods that let computers simulate weather, aircraft, and galaxies. Nearly everything in this archive that involves motion, flow, or continuous change is, underneath, a differential equation waiting to be solved.