How do I see whether or not these ODE's have a global solution:
$x'=t^2+x^2$
$x'=t^2+x$
Why?
How do I see whether or not these ODE's have a global solution:
$x'=t^2+x^2$
$x'=t^2+x$
Why?
Applying the weaker form of the Cauchy theorem (probably known to many as Picard–Lindelöf theorem) to $x'=f(t,x),$ $\partial f/\partial x$ is required to be bounded (this implies it is Lipschitz with respect to $x$, for any $t$), and this is true for the second equation where such derivatives gives $1$, but not for the first, where it gives $2x$.
Here is a paper relating to this matter: Ref.