# 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.