I'm trying to solve the following partial differential equation: $$ f(x,y) = \partial_x^2f(x,y)-\partial_y^2f(x,y) $$ for which one obvious solution would be: $$ f(x,y)=e^{2x+\sqrt{3}y} $$ An even more general solution would be: $$ f(x,y)=c_1e^{\cosh(\alpha)\,x+\sinh(\alpha)\,y}+c_2e^{-(\cosh(\alpha)\,x+\sinh(\alpha)\,y)} $$ for arbitrary $\alpha,\,c_1$ and $c_2$.
When trying to let Mathematica solve this equation via
DSolve[f[x, y] == D[f[x, y], {x, 2}] - D[f[x, y], {y, 2}], f[x, y], {x, y}]
it just returns the input. Did I set up DSolve the wrong way?