Show that the initial value problem has a solution on R.

**alphabeta89** · November 3rd 2012, 04:18 AM

Show that the initial value problem $x' = \frac{{x^3}{e^{t}}}{1+x^2} + {t^2}{\cos x}, x(0) = 1$ has a solution on $\mathbb{R}$ .

I tried to use Picard's theorem, but to no avail I can't get any result.

**chiro** · November 3rd 2012, 04:30 AM

Hey alphabeta89.

It's been a while since I did DE's, but have you heard of the Lipschitz condition for the existence of a solution for a particular DE?

**alphabeta89** · November 3rd 2012, 04:41 AM

Nope, how do I go about doing it?

**chiro** · November 3rd 2012, 03:05 PM

I haven't watched the video myself, but you might want to look at this:

**alphabeta89** · November 3rd 2012, 04:01 PM

Ok, so which domain of t and x do I apply on?

**chiro** · November 3rd 2012, 04:06 PM

Well you have a positive period of L so you can think about the circle going from 0 to 2pi and the other object going from 0 to L.

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