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Using Gronwall Inequality to Prove Uniqueness/Existence

  1. Sep 23, 2012 #1

    I need to use the Gronwall inequality to discuss existence/uniqueness of the solution to the initial value problem:

    x'(t)=xsin(tx) + t with initial condition x(t0) = x0.

    I can convert this into integral form

    [tex]x(t) = x_0 + \int\limits_{t_0}^{t} xsin(sx) + s ds[/tex]

    Which of course can reduce to

    [tex]x(t) = x_0 + \frac{t^2}{2} - \frac{t_0^2}{2} + \int\limits_{t_0}^{t} xsin(sx) ds[/tex]

    But I'm unsure of how to progress from here. Also any links to other good places to study this material would help.

    Thank you.
    Last edited: Sep 23, 2012
  2. jcsd
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