Find the general solution of the following differential equation: $x \frac{dy}{dx} = y - x \sin\left(\frac{y}{x}\right)$

$x\left(\text{cosec } \frac{y}{x} - \cot \frac{y}{x}\right) = C$

The correct answer is Option (2) → $x\left(\text{cosec } \frac{y}{x} - \cot \frac{y}{x}\right) = C$ ##

We have the differential equation:

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

The equation is a homogeneous differential equation.

Putting $y = vx ⇒\frac{dy}{dx} = v + x \frac{dv}{dx}$

The differential equation becomes:

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

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

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

$\Rightarrow \text{cosec } v \, dv = -\frac{dx}{x}$

Integrating both sides, we get:

$\log |\text{cosec } v - \cot v| = -\log |x| + \log k, \quad k > 0$

(Here, $\log |k|$ is an arbitrary constant)

$\Rightarrow \log |(\text{cosec } v - \cot v)x| = \log k$

$\Rightarrow |(\text{cosec } v - \cot v)x| = k$

$\Rightarrow x(\text{cosec } v - \cot v) = \pm k$

$\Rightarrow x\left(\text{cosec } \frac{y}{x} - \cot \frac{y}{x}\right) = C$

which is the required general solution.