# Global existence of solutions

How do I see whether or not these ODE's have a global solution:

$$x'=t^2+x^2$$

$$x'=t^2+x$$

Why?

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## 2 Answers

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 $2x$.

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Here is a paper relating to this matter: Ref.

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