# nLab ordinary differential equation

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

An ordinary differential equation is a differential equation involving derivatives of a function with respect to one argument only, i.e. the function is on a manifold only of dimension $d = 1$. This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter.

## Existence and uniqueness (Picard-Lindelöf theorem)

A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let $X$ be a Banach space; denote the ball of radius $r$ about $x \in X$ by $B_r(x)$. Given $(t_0, x_0) \in \mathbb{R} \times X$ and $a, r \gt 0$, put $Q(a, r) \coloneqq [t_0 - a, t_0 + a] \times B_r(x_0)$.

###### Theorem

(Picard-Lindelöf) Suppose $f: Q(a, b) \to X$ is a continuous function satisfying the following conditions:

• (Lipschitz condition) There is a Lipschitz constant $L$ such that

$\|f(t, x) - f(t, x')\| \leq L\|x - x'\|$

for all $(t, x) \in Q(a, b)$;

• (Boundedness) There is a constant $K$ such that $\sup_{(t, x) \in Q(a, b)} \|f(t, x)\| \leq K$.

Then for any $y_0 \in X$ and $c \coloneqq \min(a, b/K)$, there exists exactly one solution $y: [t_0 - c, t_0 + c] \to X$ to the initial value problem

$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.$
###### Proof

Revised on April 30, 2015 19:04:18 by Urs Schreiber (195.113.30.252)