Question

I have the following diff eq:
xdy + (xy+y-1)dx =0
My attempt to solve: Since they are not exact I tried to find the integrating factor by using
exp{S 1/M [ dN/dx - dM/dy)dy to which I got e^x.
I am supposed to multiply this integrating factor with the original eq. but this doesn't take me anywhere. I am stuck.. please help! Thanks.

Answer 1

xdy+xydx+ydx-dx=0
xydx+ydx-dx=-xdy
dx(xy+y-1)=-xdy
xy+y-1=-x(dy/dx)
(xy+y-1)/(-x)=dy/dx
-y-(y/x)+(1/x)=dy/dx

Answer 2

or if you only wanted y'
xy'+xyx'+yx'-x'=0
y'=(xyx'+yx'-x')/-x
y'=-yx'-(y/x)x'+(1/x)x'

Answer 3

well I was supposed to use an integrating factor and the convert it into an exact equation... but thanks for the help