Answer to Question #199553 in Differential Equations for Raj kumar

Question

Solve the differential equation

xcos(y/x) (ydx+xdy) = ysin(y/x) (xdy-ydx)

Answer 1

$$x\cos(\dfrac{y}{x})(ydx+xdy)=y\sin(\dfrac{y}{x})(xdy-ydx)$$

$$\dfrac{ydx+xdy}{xy}=\tan(\dfrac{y}{x})\dfrac{xdy-ydx}{x^2}$$

$$\dfrac{d(xy)}{xy}=\tan(\dfrac{y}{x})d(\dfrac{y}{x})$$

$$\int\dfrac{d(xy)}{xy}=\int\tan(\dfrac{y}{x})d(\dfrac{y}{x})$$

$$\int \tan u du=\int\dfrac{\sin u}{\cos u}du$$

$$=-\ln|\cos u|+C_1=\ln|\sec u|+c_1$$

$$\ln|xy|=\ln|\sec (\dfrac{y}{x})|-\ln c$$

$$\sec (\dfrac{y}{x})=cxy$$