Originally Posted by
Danny If the ODE is homogeneous then it admits the symmetry
$\displaystyle
\bar{x} = e^{\varepsilon \alpha}x,\;\;\;\bar{y} = e^{\varepsilon \alpha}y,
$
or infinitesimal transformation
$\displaystyle
\bar{x} = x + \varspsilon \alpha x + O(\varepsilon^2),\;\;\;
\bar{y} = y + \varspsilon \alpha y + O(\varepsilon^2)
$
and the original ODE has the integrating factor
$\displaystyle
\mu = \dfrac{1}{\alpha x M + \alpha y N}
$ (we can set $\displaystyle \alpha = 1$ wlog)
This was show by Lie in the late 1800's.