The following PDE came up while working on a problem.
$$ f_{xx}f_{yy} - f_{xy}^2=-1 $$ $$f(x,0) = 0$$ $$f_y(x,0)= \frac{1}{x}$$
Separation of variables did not work and I am still looking for at least one solution to the PDE. The variable $x$ is taken to be greater than $1$, and the variable $y$ is taken to be nonnegative.