Maximal solutions of $x' = \sqrt{|x|}$, $x(0) = 0$

Question

Given the initial value problem, $x' = \sqrt{|x|},x(0) = 0$ I need to determine all its solutions.

Clearly, $\phi_1(t) = 0,\phi_2(t) = \frac{1}{4} t|t|,\phi_3(t) = \frac{1}{4}(t)_{+}^2$ are solutions in $\mathbb{R}$. Now my question is how can I determine all maximal solutions of the problem.

Edit

Remembering my rusted differential equations now I know better where these solutions come from, one could use the method of separable equations.

Furthermore, the shown solutions have been built by "concatenating" basic solutions $0$, $\mathbb{R} \to \mathbb{R}$, $\frac{1}{4}t^2$, $]0,\infty[ \to \mathbb{R}$ and $\frac{-1}{4}t^2$. There is a general theorem that allows to do this when any two solutions coincide in a point of the interval.

Still, the question remains open. Does the method of separable variables give all the solutions in a given interval? If so, is concatenation the only procedure to compose to solutions?

Answer 1

For every $t_-\leqslant0\leqslant t_+$, finite or infinite, the function $$x(t)=-\frac14(t-t_-)^2\mathbf 1_{t<t_-}+\frac14(t-t_+)^2\mathbf 1_{t>t_+}$$ solves this differential equation with the initial condition $x(0)=0$. You might want to show these are all the solutions.