Chapter 9: First-Order Differential Equations - Section 9.3 - Applications - Exercises 9.3 - Page 544: 5

$x^2+y^2=K$

We have the differential equation: $-\dfrac{dx}{dy}=\dfrac{y}{x}$ $\implies -x \ dx =y \ dy$ Now, we will integrate it. $-\int x dx=\int y dy$ $\implies -\dfrac{x^2}{2}=\dfrac{y^2}{2}+C$ $\implies \dfrac{-x^2-y^2}{2}=C$ $\implies x^2+y^2=-2C$ This can be re-written as: $x^2+y^2=K$