Answer to Question #219482 in Differential Equations for Sam

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

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