Solution of an ordinary differential equation

Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an ordinary differential equation of order $n$, with $F$ defined on a set $U \subseteq \mathbb{R}^{n+2}$. Let $I$ be an open interval of $\mathbb{R}$. A solution of the equation on $I$ is a function $u : I \to \mathbb{R}$ with these two properties. The function $u$ is $n$ times differentiable on $I$, and $\left(x, u(x), u'(x), \dots, u^{(n)}(x)\right) \in U$ for each $x \in I$. For each $x \in I$, $F\left(x, u(x), u'(x), \dots, u^{(n)}(x)\right) = 0$. For a system, $u$ takes values in $\mathbb{R}^m$ and the same two conditions hold.