Came across this problem in exam revision which seems different to the usual integral equation/contraction mapping question I've done,
Let $y(t) \in C^1[0,\infty)$ satisfy $$y'(t) = 2\sqrt{\left|y(t)\right|} \ \ \text{for } t>0 ,\quad y(0) = 0$$
Give a detailed proof showing that there exists $a \in [0,\infty ) $ such that $$y(t) = \begin{cases}0 \ \text{ if } \quad 0 \leq t \leq a \\ (t-a)^2 \ \text{if} \quad t > a \end{cases}$$
So I know that I can recast the problem as some integral equation over a bounded domain, which in this case will depend on $a$, but I'm having difficulty even recognizing what that will be.
Thanks.
Actually that's not quite true. The solution could also be $y(t) = 0$ for $0 \le t < \infty$.
The differential equation is separable. In the region $y(t) > 0$ the Existence and Uniqueness Theorem applies, and the solutions are all of the form $y(t) = (t - a)^2$ for $t > a$. Similarly, a solution in the region $y < 0$ would be of the form $y(t) = - (t-b)^2$ for $t < b$, but this can't fit with $y(0)=0$ and $t > 0$.