How to use Greens functions to solve variation of Helmholtz equation !?

DoctorDeath64

Hey I'm trying to solve the following equation:

d²/dx² f(x,y) - d²/dy² f(x,y) + λf(x,y)=δ(x)δ(y) .

The right hand side seems to indicate that Greens functions will be useful here but the solution f(x,y) isn't axisymmetric when it's transformed to polars, therefore you're left with a PDE. I also thought about (d/dx -d/dy)(d/dx+d/dy) and substituting in X₁=x+y and X₂=x-y, however assuming separation of variables for f(X₁,X₂)=A(X₁)B(X₂) gives:

d/dX₁ A(X₁) d/dX₂B(X₂)+ λA(X₁)B(X₂)=δ(0.5(X₁+X₂))δ(0.5(X₁-X₂))

which doesn't lend itself to be entirely separable. Any advice would be much appreciated !