I am given a differential equation and asked to find the area where it's lipschitz. $$\frac{d^3}{dt^3}u+u^2=1$$ What I did so far is that I converted the DE to a system of equations with this method: $$x_1=u$$ $$x_2=\frac{d}{dt}u$$ $$x_3=\frac{d^2}{dt^2}u$$
Then
$$X'=\begin{bmatrix}x_2\\x_3 \\1-x_1^2\end{bmatrix}=F(t,X)$$ And then we need to see where $$\frac{|F(t,X)-F(t,Y)|_3}{|X-Y|_3}$$ Is bounded. What I found after a bit of calculations is that if the domain was bounded I could find a bound for the fraction too, so if the domain is bounded then we have the lipschitz condition. But is that enough? Or should I ask if I don't know whether the domain is bounded or not how can I answer the question??