TITLE: Solve a nonlinear differential equation of the first order
QUESTION [5 upvotes]: I am trying to solve a nonlinear differential equation of the first order that comes from a geometric problem ; $$x(2x-1)y'^2-(2x-1)(2y-1)y'+y(2y-1)=0.$$
edit1   I am looking for human methods to solve the equation 
edit2  the geometric problem was discussed on this french forum http://www.les-mathematiques.net/phorum/read.php?8,1779080,1779080 
We can see the differential equation here  http://www.les-mathematiques.net/phorum/read.php?8,1779080,1780782#msg-1780782
edit 3 I do not trust formal computer programs: look at Wolfram's answer when asked to calculate the cubic root of -1 https://www.wolframalpha.com/input/?i=%7B%5Csqrt%5B3%5D%7B-1%7D%7D%29+.

REPLY [2 votes]: This problem is about isotomic transversals: 
In general, each point of the plane is on a pair of isotomic transversals with respect to some given triangle $ABC$.
So is defined a field of directions in the plane.
Write down the differential equation linked to this field and integrate it.