Let $F(x_1,x_2,u)$ be a real valued function. Consider the following PDE $$f_1(x_1,x_2,u) \left(F-u \frac{\partial F}{\partial u}\right) +f_2(x_1,x_2,u) \frac{\partial F}{\partial u} =0.$$ The solution of the above PDE is given by $$F=\phi(x_1,x_2)\exp\left( \int \frac{f_1}{u f_1 -f_2} du\right).$$
The question is: according to the above solution $F$, is it possible with out loss of genrality to write $F$ in the form $$F=\rho(x_1,x_2) f(u), \quad \text{or} \quad F=f(\rho(x_1,x_2) u)? $$