# Using Gronwall Inequality to Prove Uniqueness/Existence

1. Sep 23, 2012

### JMeyer

Hello,

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

$$x(t) = x_0 + \int\limits_{t_0}^{t} xsin(sx) + s ds$$

Which of course can reduce to

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

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