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:

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

Solving for C:

Implicit Differentiation of the original equation:

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

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.