Isolate the derivative and then put it into differential form:
$\displaystyle (y-x)y'=2y^2-2y'$
$\displaystyle y'(y-x+2)=2y^2$
$\displaystyle dy(y-x+2)=2y^2 dx$
Now, can you find an integrating factor?
I follow you till there. But whats an integrating factor? and how can you move the x's to the dx's and the y's to the dy's???? please help further! thank!!