To state it another way, does there exist a function $f(x, y, y')$ such that for each $x_0, y_0 \in \mathbb{R}^2$ the IVP $f(x, y, y') = 0; y(x_0) = y_0$ has infinitely many solutions?
So far, I have checked functions that are not Lipschitz continuous, but for each of these functions only one IVP has infinitely many solutions. E. g. $y'=y^{1/3}$ has infinitely many solutions for $y(0)=0$. And I am wondering, is there any differential equation satisfying the conditions above?