I have general question about system of partial differential equations. Conside a function $\psi = \psi(x,y)$ and a system two partial differential equtions $$ \psi_{xx} + \psi_{x} = F \tag{1} $$ $$ \psi_{yy} + \psi_{y} = G \tag{2} $$ where $\psi_z= \frac{\partial \psi}{\partial z}$ denotes the partial derivate and $F=F(x,y),G=G(x,y)$ are functions of $x,y$.
Now consider having the solution of the equation (1), let's denote the solution of equations (1) as $\psi^{(1)}$ i.e. $ \psi^{1}_{xx} + \psi^{1}_{x} = F $ holds.
The question I have is what do I plug into the equation (2). At first glance, it seems to me, that any part of $\psi$ purely dependent on on $y$ would be derivatived to zero, so that I should plug in $\psi = \psi^1 + f(y)$ so that $$ \psi_{yy} + \psi_{y} = \psi^{(1)}_{yy} + f_y + \psi^{(1)}_{yy} + f_{yy} $$ where, again $f = f(y)$. Now I would plug this into $(2)$ and try to solve it.
But this seems problematic, as if I would first solve the equation (2), the solution would have the form $\psi(x,y) = \psi^{(2)}(x,y) + f(x)$, where I denoted $\psi^{(2)}$ as the solution of equations (2). This "asymmetry" doesn't look good, since the order of solving partial differential equation should surely not impact the form of the solution of a system of equations.
So the question is, if I have solution of one of the equations (1) or (2), what should I plug in into the second unsolved equations to ensure that both equations are satisfied?
