Explanation

(c)
Given, $\frac{dy}{dx} = \frac{y\sin{\left( \frac{y}{x} \right) - x}}{x\sin\left( \frac{y}{x} \right)} = \frac{\frac{y}{x}\sin{\left( \frac{y}{x} \right) - 1}}{\sin\left( \frac{y}{x} \right)}$
Put $\frac{y}{x} = u$
$\Rightarrow$ $\frac{dy}{dx} = x\ \frac{du}{dx} + u$
$\therefore$ $x\ \frac{du}{dx} + u = \frac{u\sin{u - 1}}{\sin u}$
$\Rightarrow$ $- \sin u\ du = \frac{1}{x}\ dx$
$\Rightarrow$ $\cos u = \log x + c$ [integrating]
$\Rightarrow$ $\cos{\left( \frac{y}{x} \right) = \log x + c}$
$\therefore$ $y(1) = \frac{\pi}{2}$
$\therefore$ $\cos\frac{\pi}{2} = \log{1 + c}$
$\Rightarrow$ $c = 0$
Thus, $\cos\left( \frac{y}{x} \right) = \log x$