If you're given the arbitrary function:
$f(u(x,y),v(x,y))$
Solve for functions $u$ and $v$ that satisfy the following equation (for any function $f$).
$ \frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 f}{\partial u \ \partial v} $
I attempted to solve this by first solving for $f_{xx}$ and $f_{yy}$ then subtracting them to see if something cancels.
$ f_{xx} = f_u u_{xx} + f_v v_{xx} + f_{uu} u_x^2 + f_{vv} v_x^2 + 2f_{uv} u_x v_x $
$ f_{yy} = f_u u_{yy} + f_v v_{yy} + f_{uu} u_y^2 + f_{vv} v_y^2 + 2f_{uv} u_y v_y $
Subtracting these results in a long nonsense equation that doesn't have much meaning.
$ f_{xx} - f_{yy} = f_u (u_{xx} - u_{yy}) + f_v(v_{xx} - v_{yy}) + f_{uu} (u_x^2 - u_y^2) + f_{vv} (v_x^2 - v_y^2) + 2f_{uv} (u_x v_x - u_y v_y) $
How would I go about solving this? Is this a partial differential equation problem?