Answer to Question #188960 in Differential Equations for Shiv

Question

xdx+ydy=a²(xdy-ydx)/x²+y²

Answer 1

$xdx+ydy=\dfrac{a^2(xdy-ydx)}{x^2+y^2)}\\[9pt] d(x^2+y^2)=\dfrac{a^2\frac{(xdy-ydx)}{x^2}}{1+\frac{y^2}{x^2}}\\[9pt] d(x^2+y^2)=\dfrac{a^2d(\frac{y}{x})}{1+(\frac{y}{x})^2}\\[9pt] \text{ Integrating Both the sides}\\[9pt] \int d(x^2+y^2)=\int \dfrac{a^2d(\frac{y}{x})}{1+(\frac{y}{x})^2}\\[9pt] \Rightarrow x^2+y^2=a^2arctan(\dfrac{y}{x})+C\\[9pt] \Rightarrow x^2+y^2-a^2arctan(\dfrac{y}{x})=C$