Question

(3x-y-6)dx+(6x-2y-6)dy=0

Answer 1

this is not an exact equation...seems the solution cant be found using elementary functions
http://www.wolframalpha.com/input/?i=%283x-y-6%29dx%2B%286x-2y-6%29dy%3D0

to get started you need to reduce it down to 1-variable equation
notice both terms have a factor of "3x-y"
let u = 3x-y --> du/dx = 3- dy/dx
\[(u-6) + (2u-6)(3-\frac{du}{dx}) = 0\]
now you can try to solve for u(x) by separating variables
\[\frac{du}{dx} = \frac{7u-24}{2u-6}\]
\[\int\limits \frac{2u-6}{7u-24} du = \int\limits dx\]
\[\frac{2}{49}(7u-24)+\frac{6}{49}\ln (7u-24) = x +C\]
now you get stuck because there is no way to algebraically isolate "u"
if you could then final solution would be
\[y = 3x -u(x)\]