Consider the initial problem value $x' = x^2, x(0) = x_0 > 0$. What is the maximal value for $T_0$ (as a function of $x_0$) accordind with Picard-Lindelof Theorem and Picard-Lindelof improved theorem, respectively? what maximal value do you get from the explicit solution? (Hint: Compute $T_0$ as a function of $\delta$ and find optimal $\delta$).
this question of the book: ODE and dynamical systems, Gerald Teschl, pg 42.
I was able to calculate the solution of the initial value problem. But I do not know how to find these ranges according to these theorems.