# Globally defined and unique solutions.

1. Jun 25, 2010

### Malmstrom

Take a Cauchy problem like:

$$y'=(y^2-1)(y^2+x^2)$$
$$y(0)=y_0$$

Show that the problem has a unique maximal solution.
Show that if $$|y_0| < 1$$ the solution is globally defined on R whereas if $$y_0 > 1$$ it is not.
I'm having trouble with this type of questions: how does one prove global uniqueness? Only via the usual theorems? How does one show a solution is "globally" defined on R?

Another example is
$$y'=1/y - 1/x$$
$$y(1)=1$$
with a solution that should be globally defined on (0,+infinity).