$cy^2 = x^2 + y$
Divide by y2
$c = \dfrac{x^2}{y^2} + \dfrac{1}{y}$
$0 = \dfrac{y^2(2x~dx) - x^2(2y~dy)}{(y^2)^2} + \dfrac{-dy}{y^2}$
$0 = \dfrac{2xy^2~dx - 2x^2y~dy}{y^4} - \dfrac{dy}{y^2}$
$0 = \dfrac{2(xy~dx - x^2~dy)y}{y^4} - \dfrac{dy}{y^2}$
$0 = \dfrac{2(xy~dx - x^2~dy)}{y^3} - \dfrac{dy}{y^2}$
Multiply by y3
$0 = 2(xy~dx - x^2~dy) - y~dy$
$0 = 2xy~dx - 2x^2~dy - y~dy$
$2xy~dx - (2x^2~dy + y~dy) = 0$
$2xy~dx - (2x^2 + y)~dy = 0$ okay