# The Unapologetic Mathematician

## The Existence and Uniqueness Theorem of Ordinary Differential Equations (statement)

I have to take a little detour for now to prove an important result: the existence and uniqueness theorem of ordinary differential equations. This is one of those hard analytic nubs that differential geometry takes as a building block, but it still needs to be proven once before we can get back away from this analysis.

Anyway, we consider a continuously differentiable function $F:U\to\mathbb{R}^n$ defined on an open region $U\subseteq\mathbb{R}^n$, and the initial value problem:

$\displaystyle\begin{aligned}v'(t)&=F(v(t))\\v(0)&=a\end{aligned}$

for some fixed initial value $a\in U$. I say that there is a unique solution to this problem, in the sense that there is some interval $(-a,a)$ and a unique function $v:(-a,a)\to\mathbb{R}^n$ satisfying both conditions.

In fact, more is true: the solution varies continuously with the starting point. That is, there is an interval $I$ around $0\in\mathbb{R}$, some neighborhood $W$ of $a$ and a continuously differentiable function $\psi:I\times W\to U$ called the “flow” of the system defined by the differential equation $v'=F(v)$, which satisfies the two conditions:

$\displaystyle\begin{aligned}\frac{\partial}{\partial t}\psi(t,u)&=F(\psi(t,u))\\\psi(0,u)&=u\end{aligned}$

Then for any $w\in W$ we can get a curve $v_w:I\to U$ defined by $v_w(t)=\psi(t,w)$. The two conditions on the flow then tell us that $v_w$ is a solution of the initial value problem with initial value $w$.

This will take us a short while, but then we can put it behind us and get back to differential geometry. Incidentally, the approach I will use generally follows that of Hirsch and Smale.

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Posted by John Armstrong | Analysis, Differential Equations

## 4 Comments	»

1. [...] we can start actually closing in on a solution to our initial value problem. Recall the [...]

Pingback by | May 5, 2011 | Reply

2. [...] convergence of the Picard iteration shows the existence part of our existence and uniqueness theorem. Now we prove the uniqueness [...]

Pingback by | May 9, 2011 | Reply

3. [...] that we’ve got the existence and uniqueness of our solutions down, we have one more of our promised results: the smooth dependence of solutions on initial conditions. That is, if we use our existence and [...]

Pingback by | May 16, 2011 | Reply

4. [...] should look familiar, since they’re very similar to the conditions we wrote down for the flow of a differential [...]

Pingback by | May 28, 2011 | Reply

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## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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