Solve the following PDE using a change of variable:
$$ \alpha^2 \dfrac{\partial^2 z}{\partial x^2} - \beta^2 \dfrac{\partial^2 z}{\partial y^2} = 0$$
This is my attemp:
Let the following change of variables:
$$ \left\{ \begin{array}{ll} \phi = a x + by \\ \eta = cx + dy \end{array} \right.$$
Now I compute the derivatives according to that change:
$$\left\{ \begin{array}{ll} z_x = a z_\phi + c z_\eta \\ z_y = b z_\phi + d z_\eta \\ z_{xx} = a^2 z_{\phi \phi} + 2 a c z_{\phi \eta} + c^2 z_{\eta \eta} \\ z_{yy} = b^2 z_{\phi \phi} + 2 b d z_{\phi \eta} + d^2 z_{\eta \eta} \end{array} \right.$$
Thus the previous PDE has now been turned into
$$\alpha^2 (a^2 z_{\phi \phi} + 2 c a z_{\eta \phi}+ c^2 z_{\eta \eta}) - \beta^2 (b^2 z_{\phi \phi} + 2 b d z_{\eta \phi} + d^2 z_{\eta \eta}) = 0$$
or equivalently
$$ (\alpha^2 a^2 - \beta^2 b^2) z_{\phi \phi} + (2 c a \alpha^2 - 2 b d \beta^2)z_{\eta \phi} + (\alpha^2 c^2 - \beta^2 d^2) z_{\eta \eta} = 0$$
Now usually, in other example I am able to solve this by chosing $a,b,c,d$ such that the coefficients of $z_{\phi \phi}$ and $z_{\eta \eta}$ are both $0$. But this time that implies that my equation becomes just $0=0$
Have I done something wrong in my calculations? Should I have used another change of variables? Is there any other way (as simple as possible, as I have just started with PDEs) to solve this PDE?