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# existence and uniqueness of solution of ordinary differential equations

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

$f:E\to W.$ |

Then the ordinary differential equation defined as

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

with the initial condition

$x(0)=x_{0}$ |

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

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

such that $\dot{x}=f\circ x$ and $x(0)=x_{0}$.[HS]

# References

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

Related:

PicardsTheorem2, CauchyKowalewskiTheorem

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

## Mathematics Subject Classification

35-00*no label found*34-00

*no label found*34A12

*no label found*

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new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

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new correction: Misspelled name by DavidSteinsaltz

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new correction: underline-typo by Filipe

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new correction: cocycle pro cocyle by pahio

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier