Order of a differential equation

Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an ordinary differential equation for an unknown function $y$ of $x$. Its order is the largest integer $k$ such that the equation depends on the derivative $y^{(k)}$. For a system of equations, the order is the largest order of a derivative that occurs in the system.

Remarks

The equation $y' = x y$ has order $1$, and the equation $y'' + \sin y = 0$ has order $2$. The order does not depend on the powers of the derivatives: $(y')^3 = y$ has order $1$. An equation of order $n$ can be changed into a first-order system of $n$ equations. See reduction to a first-order system.

Explanations

L1 human

The order tells you how many times the unknown function is differentiated, at most. An equation with $y'$ only is of first order. An equation with $y''$, but no higher derivative, is of second order.