Let y be a real function of x and let y′, y″,…, y⁽ⁿ⁾ denote the first, second,… nth derivatives of y with respect to x. An ordinary differential equation (ODE for short) is an equation involving x, y, y′, y″,…. (compare partial differential equation, stochastic differential equation). x is referred to as the independent variable and y as the dependent variable. The order of the differential equation is the order n of the highest derivative y⁽ⁿ⁾ that appears.

Ordinary and partial differential equations are hugely important in mathematical models describing phenomena in the real world (see Bessel’s equation, Black-Scholes equation, calculus of variations, Euler’s equation, heat equation, Laplace’s equation, Maxwell’s equations, Navier-Stokes equations, predator-prey equations, Schrödinger’s equation, wave equation).

The problem of solving a differential equation is to find the general form of functions y whose derivatives satisfy the equation. In certain circumstances, it can be shown that a differential equation of order n has a general solution for y, involving n arbitrary constants. Such examples include linear differential equations with constant coefficients. But this is far from the general case (see Picard’s theorem). Further information, such as initial conditions or boundary conditions, may be given to specify uniquely a solution within the general solution. For example, y″– ω‎²y = 0, models simple harmonic motion, and has general solution y(x)= Acosω‎x + Bsinω‎x. If further we know y(0) = 2 and y′(0) = 1, then this specifies the solution uniquely as y(x) = 2cosω‎x +  ω‎^–1sinω‎x.

See homogeneous first-order differential equation, linear differential equations, linear first-order differential equation, Laplace transform, separable first-order differential equation.