It is well known that under suitable conditions, the symmetry of mixed second partial derivatives reads:
$$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}.$$
Now, what will happen if we try to solve the following equation when the above situation does not hold?
$$\begin{cases}\frac{\partial f}{\partial x}=p(x,y)\\\frac{\partial f}{\partial y}=q(x,y)\end{cases},$$
where $\frac{\partial p}{\partial y}\neq\frac{\partial q}{\partial x}$
I thought there will somehow be numerical solution if we use the following iteration: $f(dx,dy)\approx f(0,0)+f_{x}(0,0)dx+f_{y}(0,0)dy$.
My question is: Will the solution of this system of equation exist? In the view of Clairaut's theorem, the solution ought to be weird, but we can somehow integrate them numerically. Any help please?