I've come across a relatively simple system PDE, now I don't want to go into the specifics but rather ask some more general questions about solving PDEs.
Consider three functions $f = f(x,y), g = g(x,y), \psi=\psi(x,y)$, a constant $A \neq 1$. Now I have the system of equations $$ \begin{align} \psi_x + A \psi_y &= f \\ \tag{1} \psi_x + \psi_y &= g \end{align} $$ Where I denote $\psi_i = \frac{\partial\psi}{\partial i}$. Now If we multiply the second equations by $-1$ and sum the equations we get $$ \psi_y = \frac{f-g}{A-1} \tag{3} $$ Now we can integrate this. $$ \psi = \int \frac{f-g}{A-1}dy + X $$ where I added a function $X=X(x)$, which would be derivated to zero in equation (3). Now we just plug in into the equation (1) to find $X(x)$ we get
$$ \psi = \frac{\int(f-g) \ dy}{A-1}+\frac{\int \left[ \left(A g-f\right)\int \left(-f_x+g_x\right) \ dy\ \right] dx}{A-1} $$
Now I wanted to ask if it can be shown if that such solution always solve the equations (1)? If so, that does mean that solutions to equations of type (1) always exist since we can find it "explicitly"? On the other hand, if such solution were not to exist what step would of finding the solution would be wrong?
Edit: As noted in the comments we should check if $\psi_{xy}=\psi_{yx}$ otherwise we would repeat the procedure, but instead first integrate over $x$ and then $y$ thus resulting in a different solution.