I was looking at the dynamics of the real solutions close to $x(0)=0$ for the non-autonomous ODE
\begin{equation} \dot{x}= \sqrt{x} +f(t) \end{equation}
where $f(t)>0$ is `small' for $t \ll 1$ (in the ODE $t$ is the independent variable). Especially the possible non-uniqueness of solutions seems interesting. I am guessing that these type of equations have a name, but I was not able to find it. Does anybody know more about these type of functions.
Remark: After some transformation you can actually rewrite the ODE in the form $(\dot{y}-1)y=f(t)$. These type of equations must have some name...
(Note that for the uniqueness theorem for first order ODEs you need smoothness (or lipschitzness) of the right hand-side w.r.t. $x$ which here clearly is not the case)