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## First order nonlinear differential equation

1. The problem statement, all variables and given/known data
Find the orthogonal trajectories of the given families of curves.
$$x^2 + y^2+2Cy=1$$

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
$$2x+2y y' + 2C y' = 0$$
Solving for C in the original equation:
$$C = \frac{1-x^2-y^2}{2y}$$
Solving for y' and substituting C:
$$y' = \frac{-2xy}{1+y^2-x^2}$$
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
$x^2+y^2-2Cx+1=0$

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?