Find the orthogonal trajectories of all circles through the points (1,1) and (-1,-1)

I believe I have reduced the problem to finding solutions to the following differential equation:

$\displaystyle y'=\frac{y^2-2xy-x^2+2}{y^2+2xy-x^2-2}$

The equation for all circles which pass through the given points is:

$\displaystyle (x+C)^2+(y-C)^2=2+2C^2$

Solving for C:

$\displaystyle C = \frac{-x^2-y^2+2}{2 (x-y)}$

Implicit Differentiation of the original equation:

$\displaystyle 2(x+C) + 2(y-C)y'=0$

$\displaystyle y'=\frac{x+C}{y-C}$

So, substituting C and taking the negative reciprocal, orthogonal trajectories must satisfy:

$\displaystyle y'=\frac{y^2-2xy-x^2+2}{y^2+2xy-x^2-2}$

And now I'm stuck. The answer in the book is the solution to this differential equation, but I have no idea how to get there from here aside from guessing the answer out of thin air.