Chapter 9: First-Order Differential Equations - Section 9.3 - Applications - Exercises 9.3 - Page 544: 6

$x^2+2y^2=K$

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