Differential Equation $y(x)'=(y(x)+x)/(y(x)-x)$ can someone give me some tips on how to solve this differential equation.
I looked at the Wolfram solution which substituted $y(x)=xv(x)$. I'd know how to solve from there, but I have know idea why they did it in the first place, well why the algorithm did it in the first place. When do you substitue $y(x)=xv(x)$? 
What are other ways on solving it?

Let's assume a equation scaling $x \to \alpha x$ and $y \to \beta y$. Under such scaling the equation becomes
$$
y'
=
{\alpha \over \beta}\,
{y + \left(\alpha/\beta\right)x \over y - \left(\alpha/\beta\right)x}
$$
Then, we can see the equation doesn't change its form whenever $\alpha = \beta$. It means $y/x$ doesn't change its form either. Then, a change of variables
$\left(~y \to {\rm f}~\right)$ like $y/x = {\rm f}\left(x\right)$ should simplify the original equation as other people ( see @Amzoti ) already showed.
For this simple equation, this analysis can be too pretentious. However, it illustrates a general technique that can be quite useful in more complicated cases.
See, for example,
Applications of Lie Groups to Difference Equations ( Differential and Integral Equations and Their Applications ) 
by Vladimir Dorodnitsyn.
