Your function is not globally Lipschitz, but it is

*locally *Lipschitz, and that's enough for Cauchy-Lipschitz theorem (for the unicity, at least:

**local unicity implies global unicity**).

Either you use the mean value theorem on a compact (

is continuously differentiable), or you perform a more explicit proof using Danny Arrigo's formula.

Note that

is a solution (and similarly with

and

). Using the local unicity (granted by Cauchy-Lipschitz), you can see that the graphs of the solutions can't intersect. Therefore,...

(edit: I've just had a look at the wikipedia, and Cauchy-Lipschitz theorem is just another name for Picard-Lindelöf theorem)