Find the solution of the following differential equation $\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y$ $$ d x=\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x d y $$

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$\begin{aligned} &\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y d x \ &\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x d y \ & \Rightarrow \quad \frac{d y}{d x}=\left[\frac{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)}{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)}\right] \cdot \frac{y}{x} \end{aligned}$ Let $\quad \frac{y}{x}=V \Rightarrow \frac{d y}{d x}=V+x \frac{d V}{d x}$ $\begin{aligned} & \Rightarrow \quad V+x \frac{d V}{d x}=\left(\frac{\frac{1}{V} \cos V+\sin V}{\sin V-\frac{1}{V} \cos V}\right) \cdot V \ & \Rightarrow \quad V+x \cdot \frac{d V}{d x}=\left(\frac{\cos V+V \sin V}{V \sin V-\cos V}\right) V \ & \Rightarrow \quad x \cdot \frac{d V}{d x}=\left(\frac{V \cos V+V^2 \sin V-V^2 \sin V+V \cos V}{V \sin V-\cos V}\right) \ & \Rightarrow \quad x \cdot \frac{d V}{d x}=\frac{2 V \cos V}{V \sin V-\cos V} \ & \Rightarrow \quad \frac{(V \sin V-\cos V)}{V \cos V} d V=2 \cdot \frac{d x}{x} \end{aligned}$ Let $V \cos V=t \Rightarrow(\cos V-V \sin V) d V=d t$ $\begin{array}{lc} \Rightarrow & \int-\frac{d t}{t}=2 \int \frac{d x}{x} \ \Rightarrow & 2 \ln x+c=-\ln t \ \Rightarrow & 2 \ln x+c=-\ln \left(\frac{y}{x} \cos \frac{y}{x}\right) \ \Rightarrow & \ln \left(x^2 \cdot \frac{y}{x} \cos \frac{y}{x}\right)=-C \ \Rightarrow & \ln \left(x y \cos \frac{y}{x}\right)=-C \ \Rightarrow & x y \cdot \cos \frac{y}{x}= \pm e^{-C} \end{array}$

Find the solution of the following differential equation \(\left\{x \cos \left(\frac{y}{x}\right)+y \sin…

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