I have a homework problem in which I wish to show that the family of curves given by
$x^2 + y^2 = c x$
where $c$ is an abitrary constant may be described by the differential equation
$\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$
I thought that I could use implicit differentiation to differentiate the original equation to get the second equation, but instead I get the equation
$\frac{c-2x}{2y}=\frac{dy}{dx}$
As you can see the derivative I get is not in the form of the equation that I am supposed to get. I do not see a way for my solution to even become similar to the proposed solution as one contains constants whereas the other does not.
What is the correct procedure I should use to solve the problem?