I think that the best thing we can do is to separate the variables.
dy/dx = 7x^2/y^3
(S) y^3 dy = (S) 7x^2 dx
(1/4) y^4 = (7/3)x^3 + C
y^4 = (28/3)x^3 + C
y = +- 4root[(28/3)x^3 + C]
e^(y^4) = e^[(28/3)x^3 + C]
e^4y = e^[(28/3)x^3] * e^C ; e^C is a constant.
Now in both sides we get
4y = C * (28/3)x^3
y = C * 28x^3/12
y = C * (7x^3)/3
I think this can help you
