It is part of a bigger problem. I thought I was just getting the function recovery wrong, today is the first time I tried such a thing.

The problem is to find the Orthogonal Trajectories of the family of curves $\displaystyle x^2+(y-c)^2=1+c^2$, c is a constant.

I then get c by itself:

$\displaystyle u(x,y)=\frac{x^2}{y}+y-\frac{1}{y}=2c$

According to my text the Orthogonal Trajectories $\displaystyle v(x,y)=c^*$ can be obtained from:

$\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$

$\displaystyle \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Which results in my original post except I probably should have used v instead of u for the function

I do have the text book solution for this problem but it doesn't use this method. It is has me in knots as to why this won't work.