# Second order PDE Separation of variables

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?

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.
• Aren't both C and $\lambda$ undermined constants? – user247327 Jan 14 at 19:42