Solving the spherical harmonics PDE using DSolve

Question

I am trying to solve the spherical harmonics PDE in Mathemtica. My code is:

pde =  1/Sin[\[Theta]]^2 D[f[\[Theta], \[Phi]], {\[Phi], 2}] + 
Cot[\[Theta]] D[f[\[Theta], \[Phi]], \[Theta]]  + 
D[f[\[Theta], \[Phi]], {\[Theta], 2}] + c f[\[Theta], \[Phi]] == 0

DSolve[ pde, f[\[Theta], \[Phi]], {\[Theta], \[Phi]}]

DSolve seems to be unable to solve this PDE. Am I making a mistake here somewhere?

Answer 1

pde = D[f[θ, ϕ], {ϕ, 2}]/Sin[θ]^2 + Cot[θ]*D[f[θ, ϕ], θ] + D[f[θ, ϕ], {θ, 2}] + 
    λ*f[θ, ϕ] == 0

You know it is periodic in ϕ so use that by assuming a solution.

f[θ_, ϕ_] = Q[θ] Cos[m ϕ]

θeq = pde/f[θ, ϕ] // Expand
(*λ - m^2*Csc[θ]^2 + Derivative[2][Q][θ]/Q[θ] + (Cot[θ]*Derivative[1][Q][θ])/Q[θ] == 0*)

DSolve[θeq, Q[θ], θ]

f[θ, ϕ] = f[θ, ϕ] /. %[[1]]
(*Cos[m*ϕ]*(C[1]*LegendreP[(1/2)*(Sqrt[4*λ + 1] - 1), m, Cos[θ]] + 
   C[2]*LegendreQ[(1/2)*(Sqrt[4*λ + 1] - 1), m, Cos[θ]])*)

I'll let you solve for the constants, since you know what your ultimate goals and objectives for this solution are.

Mathematica is still fairly new to solving pde's so don't expect it to solve all of them without some extra work.

Note if you prefer to use E^(I m ϕ) instead of Cos[m ϕ] for ϕ dependence,θeq will be the same.