I am tasked with solving this PDE using separation of variables: ($ \alpha , \beta , \gamma $ are constants)
$$ z_{xy} + \alpha z_x + \beta z_y - \gamma z = 0 $$
By assuming $ z = X(x)Y(y) $ with separation constant $ \lambda $, I acquired ODEs:
$$ X' - \lambda X = 0 $$ $$ Y' + \frac{\alpha - \lambda \gamma}{1 + \lambda \beta}Y = 0 $$
These are first order ODEs with solutions:
$$ X = C_{1}\exp(\lambda x) $$ $$ Y = C_{2}\exp\left(\frac{\lambda \gamma - \alpha}{1 + \lambda \beta} y\right) $$
This implies that:
$$ Z = XY = C\exp\left(\lambda x + \frac{\lambda \gamma - \alpha}{1 + \lambda \beta} y\right) $$
However, a second order PDF should have two unknowns and not one. What am I doing wrong?
I think the problem is simply that it is not true, in general, that a solution to a second order PDE should have two arbitrary constants.
For an $n$-th order ODE, it is true that a solution should have $n$ arbitrary constants, but the situation is not the same for an $n$-th order PDE. See for more details the answer here.