PlanetMath: existence and uniqueness of solution of ordinary differential equations

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existence and uniqueness of solution of ordinary differential equations

(Theorem)

Let $E\subset W$ where is an open subset of which is a normed vector space, and let be a continuous differentiable map

$\displaystyle f: E \to W.$

Then the ordinary differential equation defined as

$\displaystyle \dot{x} = f(x)$

with the initial condition

$\displaystyle x(0) = x_0$

where $x_0 \in E$ has a unique solution on some interval containing zero. More specifically there exists $\alpha>0$ such that the following is a unique function

$\displaystyle x:(-\alpha,\alpha) \to E$

such that $\dot{x}=f\circ x$ and

.[HS]

References

HS: Hirsch, W. Morris, Smale, Stephen.: Differential Equations, Dynamical Systems, And Linear Algebra. Academic Press, Inc. New York, 1974.

"existence and uniqueness of solution of ordinary differential equations" is owned by Daume.

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See Also: Picard's theorem, Cauchy-Kowalewski theorem

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Cross-references: function, interval, solution, initial condition, ordinary differential equation, differentiable map, continuous, normed vector space, open subset
There are 3 references to this entry.

This is version 10 of existence and uniqueness of solution of ordinary differential equations, born on 2003-05-07, modified 2006-03-09.
Object id is 4246, canonical name is ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations.
Accessed 8493 times total.

Classification:

AMS MSC:	34-00 (Ordinary differential equations :: General reference works )
	35-00 (Partial differential equations :: General reference works )
	34A12 (Ordinary differential equations :: General theory :: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions)

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