Ordinary differential equation

Definition

Let $n \ge 1$ be an integer, let $U$ be a subset of $\mathbb{R}^{n+2}$, and let $F : U \to \mathbb{R}$ be a function. An ordinary differential equation of order $n$ is an equation $$F\left(x, y, y', \dots, y^{(n)}\right) = 0$$ for an unknown real function $y$ of one real variable $x$. Here $y^{(k)}$ is the $k$-th derivative of $y$. The equation must depend on $y^{(n)}$. If $y$ takes values in $\mathbb{R}^m$ and $F$ takes values in $\mathbb{R}^m$, the equation is a system of $m$ ordinary differential equations.

Remarks

The word ordinary means that the unknown function depends on one variable only. An equation with partial derivatives of a function of several variables is a partial differential equation. The equation is in explicit form when it is solved for the highest derivative: $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$. A solution is a function on an interval that satisfies the equation at each point of the interval.

Explanations

L1 human

An ordinary differential equation is a rule that links a function of one variable with its derivatives. For example, $y'' + y = 0$ links $y$ with its second derivative. The functions $\sin x$ and $\cos x$ satisfy it.