existence and uniqueness of solution of ordinary differential equationsExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations2003-05-07 09:33:122006-03-09 11:23:34Theoremdifferential equation% this is the default PlanetMath preamble. as your knowledge
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\makeatotherLet $E\subset W$ 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 $x(0)=x_0$.\cite{HS}
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\bibitem[HS]{HS} Hirsch, W. Morris, Smale, Stephen.: Differential Equations, Dynamical Systems, And Linear Algebra. Academic Press, Inc. New York, 1974.
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