Take a Cauchy problem like:(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Globally defined and unique solutions.

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