I have a partial differential equation in 2D Cartesian space (i.e. $x$ and $y$ are orthogonal). One solution is the modified Bessel function of the second kind, order zero $(K_0())$ with argument shown below. I'm used to seeing such Bessel functions as a solution to a single variable differential equation of slightly different form. Can anyone show how to derive this solution?
$$\frac{\partial^2 f(x,y,\eta)}{\partial x^2} + \frac{\partial^2 f(x,y,\eta)}{\partial y^2} - \eta^2f(x,y,\eta) - \delta(x-a)\delta(y-b) = 0$$
$$f(x,y,\eta) = K_0(\eta\sqrt{(x-a)^2 + (y-b)^2})$$
where $a$ and $b$ are constants, $\eta$ is an independent variable, and $\delta()$ is the Dirac-delta function.