This may seem like a strange question, but I am interested in solving $$\nabla^2 u = 0, \, r > R$$ with $u \to Ur \cos \theta$ for $r \to \infty$ and $\frac{\partial u}{\partial r} \big |_{r=R} = 0$. The solution in polar coordinates is easily found to be $$ u(r,\theta) = U \cos \theta \left(r + \frac{R^2}{r} \right). $$ Correspondingly, one can then go to rectangular coordinates to obtain $$u(x,y) = U x \left( 1 + \frac{R^2}{x^2+y^2} \right),$$ via substitution.
My question is does someone know of a way to solve this in rectangular coordinates first? I attempted separation of variables but the closest I can come to a solution is $$u(x,y) = Ux + \sum_{\lambda > 0} e^{-\lambda |x|} [A_\lambda \sin(\lambda y) + B_\lambda \cos(\lambda y)],$$ which is not at all obvious how it will get to the correct form above. Does anyone have any suggestions?
EDIT: The comments correctly suggested I should have $\lambda$ run over all possible values, so I have changed the above series to now be
$$u(x,y) = Ux + \sum_{\lambda > 0} e^{-\lambda |x|} [A_\lambda \sin(\lambda y) + B_\lambda \cos(\lambda y)] + \sum_{\lambda < 0} e^{\lambda |y|} [C_\lambda \sin(\lambda x) + D_\lambda \cos(\lambda x)]$$