If dy/dx = 7x^2/y^3 and y(3)=2, find an equation for y in terms of x

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