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Picard's Existence and uniqueness theorem proves existence of solutions to an ODE on a a particular interval, but then I believe there exists some kind of theorem that allows us to continue the solution past that interval.

Is this true? Where can I find it's proof? or can someone explain it to me?

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This question may be better suited to math.stackexchange.com –  David Roberts Jun 21 '11 at 23:27
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Here's the short version: repeat the argument with the old final time the new initial time. If the function is Lipschitz, the intervals are all the same length and you get global existence and uniqueness. If the function is only locally Lipschitz, the length of the intervals could shrink so quickly that the solution blows up in finite time. –  Aaron Hoffman Jun 21 '11 at 23:32
    
Any chance of seeing how to proof works formally? –  Al SUm Jun 22 '11 at 10:24
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closed as too localized by David Roberts, Andrew Stacey, Deane Yang, Ryan Budney, Willie Wong Jun 22 '11 at 9:20

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