If $x\frac{dy}{dx}= y (log y - log x + 1), $ then the solution of the equation is

$log \left(\frac{y}{x}\right) = Cy$

The correct answer is option (2) : $log \left(\frac{y}{x}\right) = Cy$

We have,

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

Putting $y = vx $ and $\frac{dy}{dx} = v +x \frac{dv}{dx} ,$ we get

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

$⇒\frac{1}{vlog v }dv =\frac{1}{x} dv $

$⇒log ( log v ) = log x + log C$ [On integrating]

$⇒log v = Cx$

$⇒log \left(\frac{y}{x}\right) = Cx $, which is the required solution.