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

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

Printable View

- Nov 3rd 2012, 04:18 AMalphabeta89Show that the initial value problem has a solution on R.
Show that the initial value problem $\displaystyle x' = \frac{{x^3}{e^{t}}}{1+x^2} + {t^2}{\cos x}, x(0) = 1$ has a solution on $\displaystyle \mathbb{R}$.

I tried to use Picard's theorem, but to no avail I can't get any result. (Wondering) - Nov 3rd 2012, 04:30 AMchiroRe: 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? - Nov 3rd 2012, 04:41 AMalphabeta89Re: Show that the initial value problem has a solution on R.
Nope, how do I go about doing it? :D

- Nov 3rd 2012, 03:05 PMchiroRe: 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 - Nov 3rd 2012, 04:01 PMalphabeta89Re: Show that the initial value problem has a solution on R.
Ok, so which domain of t and x do I apply on?

- Nov 3rd 2012, 04:06 PMchiroRe: 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.