## 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 defined on an open region , and the initial value problem:

for some fixed initial value . I say that there is a unique solution to this problem, in the sense that there is some interval and a unique function satisfying both conditions.

In fact, more is true: the solution varies continuously with the starting point. That is, there is an interval around , some neighborhood of and a continuously differentiable function called the “flow” of the system defined by the differential equation , which satisfies the two conditions:

Then for any we can get a curve defined by . The two conditions on the flow then tell us that is a solution of the initial value problem with initial value .

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.

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

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

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[...] 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 [...]

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

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