The differential equation $\frac{dy}{dx} + \frac{x}{y} = 0$, represents the family of curves :

$x^2 - y^2 = C$ $\frac{x}{y} = C$ $xy = C$ $x^2 + y^2 = C$

$x^2 + y^2 = C$

$\frac{dy}{dx} + \frac{x}{y} = 0$ ⇒  $\frac{dy}{dx} = \frac{-x}{y}$ $y~dy = -x~dx$ integrating both sides $\int y~dy = \int -x~dx$ ⇒  $\frac{y^2}{2} = \frac{-x^2}{2} + C$ ⇒  $x^2+y^2 = 2C$ ⇒  $x^2 + y^2 = C$