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.
Title | existence and uniqueness of solution of ordinary differential equations |
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Canonical name | ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations |
Date of creation | 2013-03-22 13:36:50 |
Last modified on | 2013-03-22 13:36:50 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 13 |
Author | Daume (40) |
Entry type | Theorem |
Classification | msc 35-00 |
Classification | msc 34-00 |
Classification | msc 34A12 |
Related topic | PicardsTheorem2 |
Related topic | CauchyKowalewskiTheorem |