# How to make sure that a solution of one differential equation satisfies another in a system of two partial differential equations.

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?

• In this particular instance, you can integrate both equations using an integrating factor \begin{align} e^{x} \psi_{x} &= f_{1}(y) + \int e^{x} F(x, y) dx & \implies \psi &= f_{2}(y) + \int e^{-x} \left( f_{1}(y) + \int e^{x} F(x, y) dx \right) dx \\ e^{y} \psi_{y} &= g_{1}(x) + \int e^{y} G(x, y) dy & \implies \psi &= g_{2}(x) + \int e^{-y} \left( g_{1}(x) + \int e^{y} G(x, y) dy \right) dy \end{align} You can then determine the conditions on $F, f_{1}, f_{2}, G, g_{1}, g_{2}$ such that the equations are satisfied. Commented Aug 15, 2023 at 1:21