Uniqueness of solution 

  
*Solve the differential equation $\dot x=1+x^2$, with $x(0) = x_0$. Why can you be certain that the solution is unique? Show that for all values of $x_0$ the solution goes to infinity in finite time, both forwards and backwards.
  

I have found the solution to be $x(t)=\tan(t+\arctan x_0)$. How would I go about proving uniqueness? Is the solution the entire graph of $\tan(t+\arctan x_0)$ or only the tan wave which is defined on the interval (when $\arctan(x_0)>0$):
$$-\frac{\pi}2-\arctan x_0<t<\frac{\pi}2-\arctan x_0   $$
i.e the tan wave with domain containing $t=0$ so that the tan wave intersects the $x$ axis at $x_0$

Uniqueness follows from the general Picard-Lindelöf (or if you prefer, Cauchy-Lipschitz) existence and uniqueness theorem for differential equations. 
Since the solution blows up as $t + \arctan(x_0)$ approaches $\pm \pi/2$, the solution is only on the interval $-\pi/2 < t +  \arctan(x_0) < \pi/2$.
