> I got lost with the words "We begin by rewriting > the equation..." > > Which equation again? Not Mike's. Not Darboux's. > > I wasn't able to find an equation in x and y only, > that's being rewritten.
I was putting Mike's equation in implicit form, after replacing f(x) by y (since we're working in the standard (x,y) coordinate plane).
y = (x-1)(x-2) / (x+1)(x+2)
y = (x^2 - 3x + 2) / (x^2 + 3x + 2)
y(x^2 + 3x + 2) = x^2 - 3x + 2
yx^2 + 3xy + 2y = x^2 - 3x + 2
yx^2 + 3xy + 2y - x^2 + 3x - 2 = 0
(yx^2 - x^2) + (3xy + 3x) + (2y - 2) = 0
(y-1)x^2 + 3(y+1)x + 2(y-1) = 0
I see now that there is a typo in my last equation quoted in your post, however. I wrote "2(x-1)", but it should have been "2(y-1)".