Second-order equation $u''_{xy}+2xyu'_y-2xu=0.$ 
solve it for $u(x,y)$.
I received the following equations:
$u=\frac{1}{2x}v'_x+yv,$
$v''_{xy}+2xyv'_y=0.$
where $v=u'_y$.
All my following tryings are worthless. I can't get the right answer, which is on this screenshot:

where $g$ and $f$ are arbitrary functions.

The formula (your screenshot) was probably obtained as shown below :

One typo corrected.
