Solving a DE using substitution method. Please Help! I'm attempting to solve the following DE:
$$y^2\frac{dy}{dx} +\frac{y^3}{x}=\frac{2}{x^2}$$
with the substitution $u(x) = y^3$
I can somewhat picture what needs to be done, but I seem to come up short.  For instance, I know the following:
$$u(x) = y^3$$
$$\frac{du}{dx}= 3y^2\frac{dy}{dx}$$
Therefore:
$$\frac{dy}{dx}= \frac{1}{3y^2}\frac{du}{dx}$$
Furthermore:
$$\frac{dy}{dx}=\frac{1}{3u^{\frac{2}{3}}}\frac{du}{dx}$$
Substituting this into the DE is where I'm getting stuck.  I can't seem to get it to "work" after that point.
I would certainly appreciate any help!

Multiply your fierst equation by 3 to egt
$$\frac{du}{dx}+\frac{3}{x}u=\frac{6}{x^2},$$
Which is a linear DE, using the integrating factor you can solve it as follows
$$\beta=\exp{\int \frac{3}{x}dx}=x^3\\
u=x^{-3}\{\int x^3*\frac{6}{x^2}dx+c\}$$ 
