Use the integrating factor μ = ex to determine the general solution (in implicit form) of the differential equation
e^y +e^{−x} ln |x| + (e^y + y^2e^−x)dy/dx=0
How to solve this?
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.