Differential equations

A differential equation is an equation that relates a function to one or more of its derivatives. Since calculus was invented partly to handle them, differential equations are the mathematical language in which the laws of physics are written: Newton's second law is a differential equation, as are Maxwell's equations, the Schrödinger equation, and the equations of general relativity. The remarkable fact is that nature, at every scale from the quantum to the cosmological, appears to be governed by local differential relations — each point in space and time determines what happens next, and global behavior emerges from the accumulation of these infinitely many local decisions.

Differential equations divide into ordinary (involving functions of a single variable) and partial (involving functions of multiple variables). The techniques for solving them form a central part of Mathematical analysis and remain active areas of research: most nonlinear differential equations cannot be solved in closed form, and the behavior of their solutions — including the possibility of chaotic dynamics — is often the deepest thing a physicist or mathematician needs to understand.

The unreasonable fact is that most of what we call scientific understanding consists of a differential equation with boundary conditions. To understand is to find the equation; to predict is to integrate it.