Well, how about using the hint? That equation is the same as

(e^y+ e^{-x}ln|x|)dx+ (e^y+ y^2e^{-x})dy= 0 and you are told that e^x is an integrating factor. That means that if you multiply by e^x on both sides,

(e^{x+y}+ ln|x|)dx+ (e^{x+y}+ y^2)dy= 0 is an "exact equation". Find a function F(x,y) such that F_{x}= e^{x+y}+ ln|x| and F_{y}= e^{x+y}+ y^2.