pde – Formal solutions of the Laplace equation

I'm trying to solve a Laplace equation in polar coordinates:

begin {equation} u_ {rr} + dfrac {1} {r} u_ {r} + dfrac {1} {r ^ {2}} u _ { theta theta} = 0, 0 r0
end {equation}

or $ f $ is a $ 2 pi $ periodic continuous function.

My attempt: using the method of separating variables $ u (r, theta) = R (r) Theta ( theta) $ we have $ dfrac { Theta} { Theta} = dfrac {-r ^ {2} R & # 39;} {R} – dfrac {rR & # 39; ;} {R} = – lambda $ is a constant. The boundary condition is now $[R'(a)+alpha R(a)] Theta ( theta) = f ( theta) $. Yes $ R (a) + alpha R (a) neq 0 $ then $ Theta (0) = Theta (2 pi) $ and we have a differential equation:

begin {matrix} Theta & # 39; + lambda Theta = 0 \ Theta (0) = Theta (2 pi) end {matrix}

However, it seems that this boundary condition is not sufficient to solve this equation. For example, when $ lambda = beta ^ {2}> $ 0 the general solution is $ Theta ( theta) = C cos { beta theta} + D sin { beta theta} $, so how can we find $ C, D $ and $ beta $? And if so $ R (a) + alpha R (a) = $ 0? Now I'm stuck, can anyone help?

Moreover, this exercise requires finding reasonable conditions for $ f $ so that the formal solution is a solution. I do not understand what it means and how can I solve this problem.