Solution to Second order PDE with elementary method Solve $ UU_{xy}-U_xU_y=0$ I know how to solve $ UU_{xy}+U_xU_y=0$ But i cant find  any trick to do with the differential operators .This is an exercise and im not interested in that specific pde but i want to know how to handle the differential operators and when can i "partial integrate" things  like in the second $$ UU_{xy}+U_xU_y=0$$ i said that this is $$ \frac {\partial U U_x}{\partial y}=0$$ So then i partial integrated by $y$  so $UU_x=g(x)$ which is $$\frac{U \partial U}{\partial x}=g(x)$$ then integrated again so I got $\frac{U^2} {2} =G(x)$ Did the same  by integrating first with $x$ so I got $U=\pm\sqrt{G(x)+F(y)}$ Cant find a way to do the same for the other one.

In this case $\dfrac{\partial}{\partial x} \dfrac{U_y}{U} = 0$, so
$U_y = g(y) U$ for some function $g(y)$.   Now $u'(y) = g(y) u(y)$ is a linear homogeneous ODE whose solutions are of the form $u(y) = c G(y)$ where $G(y)$ is one solution.  Thus $U(x,y) = F(x) G(y)$ for some functions $F$, $G$.  On the other hand, for any differentiable $F$ and $G$, $U(x,y) = F(x) G(y)$ satisfies your PDE.
