How many solutions does the differential equation $\displaystyle \frac{dy}{dx}=60(y^2)^{1/5}; \quad x>0, y(0)=0$ have?
First, $\displaystyle (y^2)^{1/5}$ is a very strange way to write $\displaystyle y^{2/5}$. Is that what you meant?
If it is, then this is a separable equation that is easily solvable: $\displaystyle \dfrac{dy}{y^{2/5}}= y^{-2/5}dy= 60 dx$.
What do you get when you integrate both sides?
Did you even read HallsofIvy's last post? You can only solve the equation through dividing, which means you're making the assumption that $\displaystyle \displaystyle \begin{align*} y \neq 0 \end{align*}$. WHAT IF IT WAS?!?!