Answer to Question #175570 in Differential Equations for Adarsh Singh

Question

Ydx+(y-x)dy

Answer 1

$$ydx+(y-x)dy=0$$

$$dx=-\dfrac{y-x}{y}dy$$

$$\dfrac{dx}{dy}=-1+\dfrac{x}{y}$$

Let $x=uy.$ Then

$$\dfrac{dx}{dy}=u+y\dfrac{u}{dy}$$

Substitute

$$u+y\dfrac{du}{dy}=-1+u$$

$$du=-\dfrac{dy}{y}$$

Integrate both sides

$$\int{du}=-\int{\dfrac{dy}{y}}$$

$$u=-\ln{|y|}+C$$

$$\dfrac{x}{y}=-\ln{|y|}+C$$