# 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

Title existence and uniqueness of solution of ordinary differential equations ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations 2013-03-22 13:36:50 2013-03-22 13:36:50 Daume (40) Daume (40) 13 Daume (40) Theorem msc 35-00 msc 34-00 msc 34A12 PicardsTheorem2 CauchyKowalewskiTheorem