Separation of variables won't work 
Find all solutions on $\mathbb{R}$ of the differential equation $y'=3|y|^{2/3}.$

Why wouldn't separation of variables method work for this differential equation? Why does the initial condition have to be nonzero?

I do not understand your statement "the separation of variables method will only work if the initial condition is nonzero".  
I have problem with the absolute value so I shall only help you solving $$y'=3 y^{2/3}$$ You can separate the variables and integrate both sides. This leads to $$y^{1/3}=x+c$$ and then $$y=(x+c)^3$$ I hope and wish this could help continuing with your problem.
