The general solution of the differential equation $x (\frac{dy}{dx}) = y + x \tan (\frac{y}{x})$ is

$\sin(\frac{y}{x})= cx$, where c is an arbitary constant

The correct answer is Option (2) → $\sin(\frac{y}{x})= cx$, where c is an arbitary constant

Given the differential equation:

$x \frac{dy}{dx} = y + x \tan\left(\frac{y}{x}\right)$

Let $v = \frac{y}{x} \Rightarrow y = vx$

Differentiate both sides:

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

Substitute into the original equation:

$x(v + x \frac{dv}{dx}) = vx + x \tan(v)$

$xv + x^2 \frac{dv}{dx} = vx + x \tan(v)$

$x^2 \frac{dv}{dx} = x \tan(v)$

$x \frac{dv}{dx} = \tan(v)$

$\frac{dv}{\tan(v)} = \frac{dx}{x}$

Use identity $\frac{1}{\tan v} = \cot v$:

$\cot v\, dv = \frac{dx}{x}$

Integrate both sides:

$\int \cot v\, dv = \int \frac{dx}{x}$

$\log|\sin v| = \log|x| + \log C$

$\sin v = Cx$

Recall $v = \frac{y}{x}$:

$\sin\left(\frac{y}{x}\right) = Cx$

This is the general solution.