**1. The problem statement, all variables and given/known data**
Find the orthogonal trajectories of the given families of curves.

[tex]x^2 + y^2+2Cy=1[/tex]

**2. Relevant equations**
The book has covered homogeneous and separable methods.

**3. The attempt at a solution**
To find the orthogonal trajectories, we simply find the curves whose tangents are perpendicular to the tangents of any curves in the original family. Implicit differentiation of the original equation yields

[tex]2x+2y y' + 2C y' = 0[/tex]

Solving for C in the original equation:

[tex]C = \frac{1-x^2-y^2}{2y}[/tex]

Solving for y' and substituting C:

[tex]y' = \frac{-2xy}{1+y^2-x^2}[/tex]

Our solutions are all curves whose tangent lines are the negative reciprocal of this, however this is not a separable or homogeneous equation as far as I can tell, so neither is its reciprocal, and thus I am stuck.

The answer, as given by the book, is

Spoiler

[itex]x^2+y^2-2Cx+1=0[/itex]

which I have verified is correct by checking that y' is the negative reciprocal of the calculated y' above, but how can this be derived without knowing the answer first?