Existence, Uniqueness & Bifurcations · Superchallenge problem

The IVP $y' = \sqrt{y} - y$, $y(0) = 0$ (for $y \ge 0$) has more than one solution: besides $y \equiv 0$, a solution may leave $0$ at any time. Among all solutions, what is the largest possible value of $y(2\ln 2)$?