For each $\alpha \gt 0,$ find all pairs $(x_0,y_0) \in \Bbb R^2$ such that the given IVP has a unique solution in the neighbourhood of $(x_0,y_0).$

Question

For each $\alpha \gt 0,$ find all pairs of $\left (x_0,y_0 \right ) \in \Bbb R^2$ such that the following initial value problem has a unique solution in the neighbourhood of $\left (x_0,y_0 \right )$ $$y' = y^{\alpha}\ ;\ y \left (x_0 \right ) = y_0.$$

How do I proceed? Any help will be appreciated.

Thanks for your valuable time for reading.