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 $y = \frac{\pi}{2}$.
Answer & explanation
Correct answer: option 1
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:
$\cos\left(\frac{\pi}{2}\right) = \log |1| - C$
$\Rightarrow 0 = 0 - C$
$⇒C = 0$
$∴\cos\left(\frac{y}{x}\right) = \log |x| \text{ is the required solution.}$