Elimination of arbitrary constant: $Y=C-\dfrac{\ln x}{x}$

Here it is.

To eliminate the constant of the equation

$$y = c - \frac{\ln x}{x}$$

Implicitly differentiating the above equation:

$$y' = 0 - d \left (\frac{\ln x}{x}\right)$$ $$y' = 0- \left( \frac{1-\ln x}{x^2}\right)$$ $$y' = \frac{-1+\ln x}{x^2}$$ $$x^2 y' = -1 + \ln x$$

Ultimately, we got a differential equation $x^2 y' = -1 + \ln x.$

Hope it helps.