It seems like the problem is asking for us to derive an
expression for the force F⃗ stated in terms of the unit
vectors i^ and j^.
The solution for this is:
U(x,y) = α (x^-2 + y^-2)
F = - dU(x,y)
F = - (dU/dx + dU/dy)
F = - (-2αx^-3 - 2αy^-3)
F = 2α (x^-3 + y^-3)
F = 2α (1/x^3 i + 1/y^3 j)
