By differentiating the equation:

$\displaystyle \int \frac{f(xy) + F(xy)}{f(xy) - F(xy)}\cdot \frac{d(xy)}{xy} + \ln \left(\frac{x}{y}\right) = c$

verify that:

$\displaystyle \frac{1}{xy\{f(xy) - F(xy)\}}$

is an integrating factor of $\displaystyle f(xy)\ y\ dx + F(xy)\ x\ dy = 0$