Your function has a square root term of negative value:
In[182]:= Clear["Global`*"]
In[183]:= Simplify[
y[x]^3 (1 - y[x]) x^2 - (1 - y[x])^3 x^2 + 3/8 y[x] (1/2 y[x]^(-1/2))]
Out[183]=
x^2 (-1 + y[x])^3 + (3 Sqrt[y[x]])/16 - x^2 (-1 + y[x]) y[x]^3
In[184]:= s =
NDSolve[{y'[x] ==
x^2 (-1 + y[x])^3 + (3 Sqrt[Abs[y[x]]])/16 -
x^2 (-1 + y[x]) y[x]^3, y[0] == 0}, y, {x, 0.000, 1},
Method -> Automatic]
Out[184]= {{y -> InterpolatingFunction[{{0., 1.}}, <>]}}
It works if I take absolute value underneath the square root. I also used Automatic as the integration method.
