Separation of Variables: Find the potential b/w concentric hemispheres

[ligneox]

Homework Statement

The surface depicted in the image below is constructed from three parts: (1) An outer hemispherical shell of radius 𝑏; (2) an inner hemispherical shell of radius 𝑎; and (3) a flat bottom that sits in the 𝑥 − 𝑦 plane. The potential on each of the three surfaces is specified
as 𝑉1(𝑏, 𝜃) = 0, 𝑉2(𝑎, 𝜃) = 5𝑉0 cos 𝜃 sin^2𝜃, and 𝑉3(𝑟, 𝜋⁄2) = 0. Here 𝑉0 is a constant and 𝑟, 𝜃 are the usual spherical coordinates. Find the electric potential in the hemispherical shell 𝑎 ≤ 𝑟 ≤ 𝑏.

Relevant Equations

V(r,𝜃) = sum n=0 to infinity (A_n r^n + B_n/(r^(n+1))) P_n(cos𝜃)

I'm having troubles setting up this problem. I know we are to use boundary conditions to determine An and Bn since in this case (a<r<b) neither can be set to 0. I don't know how the given potentials translate into boundary conditions, especially the V3 disk.

 

[haruspex]

I don't know how the given potentials translate into boundary conditions, especially the V3 disk.

You have a generic solution in the form of a sum, and you know the functions it must equal at the given boundaries. That gives you three equations. Write them out.

 

[ligneox]

You have a generic solution in the form of a sum, and you know the functions it must equal at the given boundaries. That gives you three equations. Write them out.

Using V1 I was able to put Bn in terms of An, so the sum now looks like
V(r,𝜃) = sum n=0 to infinity A_n (r^n - b^(2n+1)/(r^(n+1))) P_n(cos𝜃)

I'm not sure how to proceed after writing out the new sum equal to V2, V3. How do I use the Legendre polynomials? for 𝜃 = pi/2 i know Pn(cos𝜃) leaves only the even terms.

I can change the V2 to 5𝑉0 cos 𝜃 (1- cos^2 𝜃).

I feel like the next step is looking at me in the face and I can't recognize it.