Question

find a 1-parameter family of solutions. Assume dy does NOT =0
(x+(y^2-xy)^(1/2))dy-ydx=0

Answer 1

Can you confirm this is right:\[(x + \sqrt{x^2 - xy} \ ) \ dy - y \ dx = 0\]

Answer 2

In your set up, imagine you have a function F(x,y) = c.

Then
\[\frac{\partial F}{\partial x} \ dx + \frac{\partial F}{\partial y} \ dy = 0\]

Answer 3

OR are you looking at the orthogonal version?

If you are looking at it in the version above, then I'm having a very hard time convincing myself that there is any such F