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A '''differential equation''' is a mathematical equation that relates a function to its derivatives. Unlike algebraic equations, which relate variables through arithmetic operations, differential equations describe how quantities change — making them the natural language of dynamics, evolution, and continuous processes.

Differential equations fall into two broad classes:

* '''Ordinary differential equations''' (ODEs) involve functions of a single variable and their derivatives. They describe systems with a finite number of degrees of freedom — pendulums, population dynamics, chemical reaction kinetics.
* '''Partial differential equations''' (PDEs) involve functions of multiple variables and partial derivatives. They describe fields, continua, and spatially distributed systems — heat flow, fluid motion, electromagnetic propagation, gravitational curvature.

The theory of differential equations is inseparable from the concept of [[Boundary Condition|boundary conditions]] and initial conditions. A differential equation alone typically admits infinitely many solutions; specifying how the system behaves at boundaries or at an initial time selects a unique solution. This interplay between local rules (the equation) and global constraints (the boundary) is the defining structure of field theories across physics and applied mathematics.

== See also ==

* [[Boundary Condition]]
* [[Field Theory]]
* [[Fluid Dynamics]]
* [[Heat Equation]]
* [[Wave Equation]]
* [[Navier-Stokes Equations]]
* [[Dynamical Systems]]
* [[Chaos Theory]]