Solve the differential equation: $x \sin\left(\frac{y}{x}\right) \frac{dy}{dx} + x - y \sin\left(\frac{y}{x}\right) = 0$. Given that $x = 1$ when $y = \frac{\pi}{2}$.

$\cos\left(\frac{y}{x}\right) = \ln|x|$

The correct answer is Option (1) → $\cos\left(\frac{y}{x}\right) = \ln|x|$ ##

Given differential equation gives:

$\frac{dy}{dx} = \frac{y \sin\left(\frac{y}{x}\right) - x}{x \sin\left(\frac{y}{x}\right)}$

Put $\frac{y}{x} = v \Rightarrow y = vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$

$∴v + x \frac{dv}{dx} = \frac{v \sin v - 1}{\sin v}$

$\Rightarrow x \frac{dv}{dx} = \frac{-1}{\sin v}$

$\Rightarrow \int \sin v \, dv = \int \frac{-1}{x} \, dx$

$\Rightarrow -\cos v = -\log |x| + C$

$∴\cos\left(\frac{y}{x}\right) = \log |x| - C$

Given $x = 1$ when $y = \frac{\pi}{2}$, then:

$\cos\left(\frac{\pi}{2}\right) = \log |1| - C$

$\Rightarrow 0 = 0 - C$

$⇒C = 0$

$∴\cos\left(\frac{y}{x}\right) = \log |x| \text{ is the required solution.}$