# Differential Equation

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/differential-equation/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
An equation relating a function to its own derivatives. Primary mathematical language for physical laws describing rates of change.

## ODE vs PDE
Ordinary (single variable) vs Partial (multiple variables, e.g. space+time).

## Separable equations
dy/dx=ky → dy/y=k dx → y=Ae^(kx) — underlies exponential growth/decay.

## Existence and uniqueness
Picard-Lindelöf theorem: guarantees unique local solution under Lipschitz continuity condition.

## Numerical methods
Euler's method (1768) — simplest step-by-step approximation. Runge-Kutta methods — much higher accuracy.

## Chaos
Lorenz (1963): simplified atmospheric model shows sensitive dependence on initial conditions — the "butterfly effect."

## Navier-Stokes Millennium Problem
Does a smooth solution always exist for 3D fluid flow? Unsolved as of 2026. $1M Clay Prize.

## Sources
### Tier 1
- Boyce, W.E. and DiPrima, R.C. (2012). *Elementary Differential Equations*. 10th ed.
- Strogatz, S.H. (2015). *Nonlinear Dynamics and Chaos*. 2nd ed.
### Tier 2
- Fefferman, C.L. (2000). Clay Mathematics Institute Millennium Problem description.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*