A differential equation becomes ordinary when the unknown function depends on one independent variable. This restriction turns questions about motion, growth, decay, oscillation, and flow into questions about curves. The central problem is not only to solve an equation, but to understand when the equation determines a curve, how long that curve exists, how it changes when the initial state changes, and what qualitative behaviour is forced by the vector field.

The first warning is that integration alone does not explain ordinary differential equations. A formula such as $\dot{x}=f(t)$ can be solved by taking an antiderivative, but a law such as $\dot{x}=x^2$ asks the current state to determine its own future rate of change. The solution is a trajectory, and the equation can break down in finite time even when the velocity rule is smooth.

[example: Finite-Time Blow-Up for $\dot{x}=x^2$] Let $x:(-\infty,1)\to \mathbb{R}$ be defined by \begin{align*} x(t)=\frac{1}{1-t}=(1-t)^{-1}. \end{align*} At the initial time, \begin{align*} x(0)=\frac{1}{1-0}=1. \end{align*} For every $t<1$, differentiating $(1-t)^{-1}$ by the chain rule gives \begin{align*} \dot{x}(t)=(-1)(1-t)^{-2}\cdot(-1)=(1-t)^{-2}. \end{align*} Since \begin{align*} x(t)^2=\left(\frac{1}{1-t}\right)^2=\frac{1}{(1-t)^2}=(1-t)^{-2}, \end{align*} we have $\dot{x}(t)=x(t)^2$ for every $t\in(-\infty,1)$. Thus $x$ solves the initial value problem $\dot{x}=x^2$, $x(0)=1$, on $(-\infty,1)$. The endpoint $t=1$ is not a removable endpoint for a finite real-valued extension: if $M>0$ and $t\in(1-1/M,1)$, then $0<1-t<1/M$, so \begin{align*} x(t)=\frac{1}{1-t}>M. \end{align*} Hence $x(t)\to\infty$ as $t\uparrow 1$, so the solution exists locally but blows up in finite forward time. [/example]

This example separates three issues that are often conflated. There may be a local solution, it may be unique, and it may still fail to exist for all time. The theory of ordinary differential equations begins by making those questions precise.

## Definition

The basic object is an equation prescribing a derivative. To avoid hiding analytic assumptions in notation, we specify the interval of time, the state space, and the regularity expected of the unknown curve. This is the minimal framework in which the phrase "the derivative is determined by the present state" has a precise meaning.

[definition: Ordinary Differential Equation] Let $m,n,q\in \mathbb{N}$, let $I\subset \mathbb{R}$ be an interval, let $\Omega\subset(\mathbb{R}^n)^{m+1}$ be open, and let \begin{align*} \mathcal{F}:I\times \Omega\to \mathbb{R}^q \end{align*} be a function. An ordinary differential equation of order $m$ is an equation of the form \begin{align*} \mathcal{F}(t,x(t),\dot{x}(t),\ldots,x^{(m)}(t))=0, \end{align*} where $0$ denotes the zero vector in $\mathbb{R}^q$ and the unknown is an $m$-times differentiable curve $x:J\to \mathbb{R}^n$ on an interval $J\subset I$ depending on the single independent variable $t$, with $(x(t),\dot{x}(t),\ldots,x^{(m)}(t))\in\Omega$ wherever the equation is imposed. [/definition]

This definition captures the parent idea: ordinary differential equations are equations in one independent variable, not necessarily first-order equations. It also includes implicit equations, where the highest derivative may not be isolated. Most of the theory below is developed in first-order normal form because that is the setting in which existence, uniqueness, flows, and stability have their cleanest statements; passing from an implicit equation to normal form can fail at singular points, can introduce branches, and can change the natural domain of the problem.

## Initial Value Problems and Integral Form

The most important normal form is the one in which the highest derivative is already isolated. It is the form used for existence, uniqueness, flows, and stability.

The word ordinary records that the curve has one independent variable, usually time. The value $x(t)$ may have many components, so a single ordinary differential equation can encode a coupled system in $\mathbb{R}^n$. To turn the equation into a prediction rather than a family of possible motions, we must say where the trajectory starts.

Initial data are the part of the model that connect the differential law to a particular experiment. Two falling particles may obey the same differential equation while starting from different positions and velocities. The initial value problem packages the law and the starting state together.

[definition: Initial Value Problem] Let $I\subset \mathbb{R}$ be an interval, let $U\subset \mathbb{R}^n$ be open, let $F:I\times U\to \mathbb{R}^n$, let $t_0\in I$, and let $x_0\in U$. The initial value problem for $F$ with initial data $(t_0,x_0)$ is the system consisting of \begin{align*} \dot{x}(t)=F(t,x(t)) \end{align*} and \begin{align*} x(t_0)=x_0, \end{align*} for an unknown curve $x:J\to U$ on an interval $J\subset I$ with $t_0\in J$, differentiable at interior points of $J$ and one-sided differentiable at endpoints of $J$ when endpoint derivatives are used. [/definition]

The interval is part of the data of a solution, not an afterthought. A curve that satisfies the equation before a singular time and a curve that can be continued through that time have different mathematical content. The next definition records what it means to solve the problem on a specified interval while remaining inside the state space.

[definition: Solution of an Initial Value Problem] Let $I\subset \mathbb{R}$ be an interval, let $U\subset \mathbb{R}^n$ be open, let $F:I\times U\to \mathbb{R}^n$, and let $(t_0,x_0)\in I\times U$. A solution of the initial value problem is a curve $x:J\to U$, where $J\subset I$ is an interval with $t_0\in J$, such that $x$ is differentiable at every interior point of $J$ and one-sided differentiable at endpoints of $J$ when those endpoints belong to $J$, \begin{align*} x(t_0)=x_0 \end{align*} and \begin{align*} \dot{x}(t)=F(t,x(t)) \quad \text{for every } t\in J \text{ where the derivative is defined as above.} \end{align*} [/definition]

Throughout this page, $B(x_0,r)$ denotes the open Euclidean ball $\{x\in\mathbb{R}^n:|x-x_0|<r\}$, $\overline{B}(x_0,r)$ denotes the closed Euclidean ball $\{x\in\mathbb{R}^n:|x-x_0|\le r\}$, and $\mathcal{L}^1$ denotes [Lebesgue measure](/page/Lebesgue%20Measure) on the time line. In matrix formulas, $I_n$ denotes the $n\times n$ identity matrix, and $I$ denotes the same matrix when the dimension is fixed by context. These conventions let the integral and neighbourhood statements below keep their formulas short.

The differential definition is the right conceptual object, but it is not always the right analytic object. To prove existence, compare two possible solutions, or apply fixed point methods, we need an equivalent formulation involving only integration of the vector field along the unknown curve.

[quotetheorem:8311]

The integral form explains why continuity of $F$ is enough to ask for existence: integrating a continuous expression produces a differentiable curve. Uniqueness, however, needs more than continuity, and that distinction becomes visible once equations are treated as systems.

## First-Order Systems and Reduction of Order