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!

Hint: By what you wrote, we have
$$\frac{1}{3}\frac{du}{dx}+\frac{u}{x}=\frac{2}{x^2},$$
or equivalently
$$\frac{du}{dx}+\frac{3}{x}u=\frac{6}{x^2}.\tag{1}$$
You will not have any trouble with the equation $\frac{du}{dx}+\frac{3}{x}u=0$. You will also need a particular solution of (1). 
