$y = \frac{1}{2}x \ln(x) - \frac{1}{4}x + \frac{C}{x}$

$y = x \ln(x) - x + C$

$y = \frac{1}{2} \ln(x)^2 + \frac{C}{x}$

$y = \frac{\ln(x)}{x} + C x$