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.