Pve that in polar coordinates laplace equation is part2u

Question

Consider Laplace equation,

(∂²u / ∂x²) + (∂²u / ∂y²) = 0,

over the annulus in R², {(r, θ) | 1 < r < 2, -Π ≤ θ < Π} in the standard polar coordinates.

a. Prove that in polar coordinates, Laplace equation is

(∂²u / ∂r²) + (∂u / r ∂r) + (∂²u / r²∂θ²) = 0,

b. Let u(r, θ) = R(r) - Θ(θ) and derive the corresponding ODEs for R(r) and Θ(θ).

c. Assume the boundary conditions on u,

u(r, -Π)= u(r, Π), and (∂u / ∂θ) (r, -Π) = (∂u / ∂θ) (r, Π)

Find the corresponding conditions on Θ(θ). Then solve for Θ and R, and write down the general solution of u(r, θ) in an infinite series.

d. Further assume boundary conditions u(1, θ) = 0 and u(2, θ) = cos θ. Find the solution u(r, θ).