The Unapologetic Mathematician

Mathematics for the interested outsider

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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May 4, 2011 - Posted by | Analysis, Differential Equations

4 Comments »

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

    Pingback by The Picard Iteration « The Unapologetic Mathematician | 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 Uniqueness of Solutions to Differential Equations « The Unapologetic Mathematician | 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 Smooth Dependence on Initial Conditions « The Unapologetic Mathematician | 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 Integral Curves and Local Flows « The Unapologetic Mathematician | May 28, 2011 | Reply


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