Is it possible to find a solution to this differential equation? Suppose that $f_1(x,y), f_2(x,y), f_3(x,y)$ are known real valued functions. 
I am wondering if it is always possible to find another function $A(x,y)$ that satisfies:
$f_1 \frac{\partial A}{\partial x} + f_2 \frac{\partial A}{\partial y} = f_3$.
I am not even sure how to begin here, how should I proceed? Note that it isn't so important to actually find the function $A$, it would be enough to show that it exists.
 A: It is an interesting question and  I can only risk an answer. 
I believe the answer is negative, even in the case $f_3 = 0$.
Arguing by contradiction, if a solution $A$ (different than the trivial $A=c$, $c$ being a constant), to the equation 
$$f_1 \frac{\partial A }{\partial x} + f_2\frac{\partial A}{\partial y} = 0 $$ could always be found, this would imply that any dynamical system of the form 
$$ \dot{x} = f_1(x,y)$$
$$ \dot{y} = f_2(x,y)$$ admits a conserved quantity, which is I believe not true (will do thorough checking on this).
For example,  maybe too trivial, a system such as 
$$ \dot{x} = f_1(x,y)$$
$$ \dot{y} = 0$$ whose trajectories are always parallel to the x-axis. No function $A(x,y)$ different from a constant can be found to be conserved along trajectories.
A: Well, there are several ways you can solve this kind of problem. This is a PDE (partial differential equation). 
Unless you have a specific proble, I would suggest to look at the Method of Characteristics which will turn your problem to an ODE (oridnary differential equation).
However, this is just one (and quite general) method. Other method which may be simpler exist. It all depends on your choice of functions.
