Can a differential equation have non unique solutions?

Consider for example the equation $x' = 2\sqrt{|x|}$. For every $a$, the function $$ x_a(t) = \begin{cases} 0 & t < a \\ (t-a)^2 & t \ge a \end{cases} $$ is a solution. Note that for $a \ge 0$ all $x_a$ have $x_a(0) = 0$, so they are all solutions to the IVP $x' = 2\sqrt{|x|}, x(0) = 0$ and you usually discuss uniqueness for initial value problems, as otherwise uniqueness will almost never hold ($x' = 0$ has all constants as solutions).

Let your ODE be $y'-x\sqrt{y}=0, \; y(0)=0$. It is not difficult finding its solution on $\mathbb R$. It has at least two solutions as $y=0$ and $y=\frac{x^4}{16}$ passing through the origin. Can you see why the ODE has no unique solution?