Question

Xdx+ydy=a^2.xdy-ydx/x^2+y^2

Answer 1

$$xdx+ydy=\dfrac{a^2(xdy-ydxy)}{x^2+y^2}$$

$$\dfrac{1}{2}d(x^2+y^2)=\dfrac{a^2(\dfrac{xdy-ydx}{x^2})}{\dfrac{x^2+y^2}{x^2}}$$

$$d(x^2+y^2)=\dfrac{2a^2d(\dfrac{y}{x})}{1+(\dfrac{y}{x})^2}$$

Integrate

$$x^2+y^2=2a^2\tan^{-1}(\dfrac{y}{x})+C$$

Or

$$x^2+y^2-2a^2\tan^{-1}(\dfrac{y}{x})=C$$