I have the following multivariable function:
$u = x + y + F(xy)$
where F is an arbitrary differentiable function. I would like to construct a first order PDE with u(x,y) as the following solution. Initially I thought of taking $u_x$ and $u_y$ which is easy for the $x+y$ term of $u$. However, what can I do about the $F(xy)$ term? I know that it initially turns out as a solution to characteristic equations, but how do I reverse engineer a first order PDE that has $F(xy)$ as a solution? Would it require me to arbitrarily define a function of $F(xy)$ such as $F(xy) = xy$ or $F(xy) = sin(xy)$, etc. then to take arbitrary derivatives of $u_x$ and $u_y$?