I stumble upon this one not for the first time, and it looks annoyingly simple. However, I still have no idea, how to find its general solution (except for guessing it, but I didn't succeed).
$u$ is a two variable real function, $k$ is a real constant
$$\frac{\partial^2 u}{\partial x\partial y}=ku$$ If you intend to solve it with the method of separation of variables you will obtain : $$u=C\:e^{\lambda x+\frac{k}{\lambda}y}$$ with $C$ and $\lambda$ any constants.
Of course this is not the general solution, but only particular solutions.
Any linear combination of those particular solutions is a solution of the PDE.
Thus on discret form : $$u(x,y)=\sum_\lambda C_\lambda\:e^{\lambda x+\frac{k}{\lambda}y}$$ where $C_\lambda$ are arbitrary constants.
Or on integral form : $$u(x,y)=\int C(\lambda)\:e^{\lambda x+\frac{k}{\lambda}y}d\lambda$$ where $C(\lambda)$ is an arbitrary function.
The function $C(\lambda)$ has to be determined according to some boundary conditions (which are not specified in the wording of the question).