Step 1: Showing the equation is homogeneous

The given differential equation is:

$$x \frac{dy}{dx} - y + x \sin\left(\frac{y}{x}\right) = 0$$

Rearranging, we get:

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

Since the right-hand side is a function of $\frac{y}{x}$ only, the given differential equation is homogeneous.

Step 2: Solving the homogeneous equation

Let $y = vx$, where $v$ is a function of $x$.

Then:

$$\frac{dy}{dx} = v + x\frac{dv}{dx}$$

Substituting in the equation:

$$v + x\frac{dv}{dx} = v - \sin(v)$$ $$x\frac{dv}{dx} = -\sin(v)$$ $$\frac{dv}{\sin(v)} = -\frac{dx}{x}$$ $$\text{cosec}(v) \, dv = -\frac{dx}{x}$$

Integrating both sides:

$$\int \text{cosec}(v) \, dv = -\int \frac{dx}{x}$$ $$\ln|\text{cosec}(v) - \cot(v)| = -\ln|x| + C$$ $$\ln|\text{cosec}(v) - \cot(v)| + \ln|x| = C$$ $$\ln|x(\text{cosec}(v) - \cot(v))| = C$$

Substituting back $v = \frac{y}{x}$:

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

∴ This is the required general solution.