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(C) Given the differential equation: $\{x \cos (y/x) + y \sin (y/x)\} y dx = \{y \sin (y/x) - x \cos (y/x)\} x dy$. Rearranging the terms,we get: $\fr

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$xy \cos (y/x) = \pm e^{-c}$

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(C) Given the differential equation: $\{x \cos (y/x) + y \sin (y/x)\} y dx = \{y \sin (y/x) - x \cos (y/x)\} x dy$. Rearranging the terms,we get: $\frac{dy}{dx} = \frac{y \{x \cos (y/x) + y \sin (y/x)\}}{x \{y \sin (y/x) - x \cos (y/x)\}} = \frac{(y/x) \cos (y/x) + (y/x)^2 \sin (y/x)}{(y/x) \sin (y/x) - \cos (y/x)}$. Let $v = y/x$,so $y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$. Substituting these into the equation: $v + x \frac{dv}{dx} = \frac{v \cos v + v^2 \sin v}{v \sin v - \cos v}$. $x \frac{dv}{dx} = \frac{v \cos v + v^2 \sin v - v(v \sin v - \cos v)}{v \sin v - \cos v} = \frac{v \cos v + v^2 \sin v - v^2 \sin v + v \cos v}{v \sin v - \cos v} = \frac{2v \cos v}{v \sin v - \cos v}$. Separating the variables: $\frac{v \sin v - \cos v}{v \cos v} dv = 2 \frac{dx}{x}$. Integrating both sides: $\int (	an v - \frac{1}{v}) dv = 2 \int \frac{1}{x} dx$. $-\ln |\cos v| - \ln |v| = 2 \ln |x| + C$. $-\ln |v \cos v| = \ln |x^2| + C$. $-\ln |(y/x) \cos (y/x)| = \ln |x^2| + C$. $\ln |(y/x) \cos (y/x)| + \ln |x^2| = -C$. $\ln |xy \cos (y/x)| = -C$. $xy \cos (y/x) = \pm e^{-C}$.

Find the solution of the following differential equation: $\{x \cos (y/x) + y \sin (y/x)\} y dx = \{y \sin (y/x) - x \cos (y/x)\} x dy$.

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