Step 1: Rewrite the given equation:

$x \frac{dy}{dx} + y - x + xy \cot x = 0$

Dividing by $x$ (since $x \neq 0$):

$\frac{dy}{dx} + \frac{y}{x} + y \cot x = 1$

Step 2: Rearrange to standard linear form:

$\frac{dy}{dx} + y\left(\frac{1}{x} + \cot x\right) = 1$

Here $P = \frac{1}{x} + \cot x$ and $Q = 1$

Step 3: Calculate the Integrating Factor (I.F.):

$I.F. = e^{\int P \, dx} = e^{\int \left(\frac{1}{x} + \cot x\right) dx} = e^{\log|x| + \log|\sin x|} = e^{\log|x \sin x|} = x \sin x$

Step 4: Apply the general solution formula:

$y \times I.F. = \int Q \times I.F. \, dx + C$

Step 5: Calculate:

$y \cdot x \sin x = \int x \sin x \, dx + C$

Using integration by parts:

$\int x \sin x \, dx = -x \cos x + \sin x$

Step 6: Therefore, the general solution is:

$y \cdot x \sin x = -x \cos x + \sin x + C$

or

$y = -\cot x + \frac{1}{x} + \frac{C}{x \sin x}$