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Math Help - Show solutions are global

  1. #1
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    Show solutions are global

    I need to show that al the solutions of x' = x + tcosx are global.

    I think I understand what I have to do: I need to show that the solutions are defined on R.

    I know that I need to construct barriers/fences to show this but I'm not sure how to do this for this de.

    Thanks for any help.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Given a first order ODE...

    x^{'}= f(x,t) (1)

    ... conditioned by x(t_{0})= x_{0}, if in [x_{0},t_{0}] both f(*,*) and its partial derivative f_{x}^{'}(*,*) exist and are continous, then there is one and only one solution of (1) so that x(t_{0})=x_{0}. In the case you have proposed is...

    f(x,t)= x + t\cdot \cos x

    ... that is contionous in al the [x,t] plane, as well as...

    f_{x}^{'}(x,t)= 1 - t\cdot \sin x

    So all the solutions of...

    x^{'}= x + t\cdot \cos x (2)

    ... are global...

    Kind regards

    \chi \sigma
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