Take a Cauchy problem like: [tex] y'=(y^2-1)(y^2+x^2) [/tex] [tex] y(0)=y_0 [/tex] Show that the problem has a unique maximal solution. Show that if [tex] |y_0| < 1 [/tex] the solution is globally defined on R whereas if [tex] y_0 > 1 [/tex] 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 [tex] y'=1/y - 1/x [/tex] [tex] y(1)=1 [/tex] with a solution that should be globally defined on (0,+infinity). Any insight is helpful.