Existence, Uniqueness & Bifurcations · Challenge problem

For $y' = t^2 + y^2$, $y(0) = 0$, the Picard–Lindelöf theorem on the rectangle $|t| \le a$, $|y| \le b$ guarantees a solution on $|t| \le h$, where $h = \min\!\left(a, \dfrac{b}{M}\right)$ and $M = \max |f|$ on the rectangle. Over all choices of $a > 0$ and $b > 0$, what is the largest $h$ the theorem can guarantee?