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

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

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.

2. ## Re: Show that the initial value problem has a solution on R.

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?

3. ## Re: Show that the initial value problem has a solution on R.

Nope, how do I go about doing it?

4. ## Re: Show that the initial value problem has a solution on R.

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

What is a Lipschitz condition? - YouTube

5. ## Re: Show that the initial value problem has a solution on R.

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

6. ## Re: Show that the initial value problem has a solution on R.

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.