Question

Consider the partial differential equation

[1/r²∂/∂r r² ∂/∂r + 1/r²sin²θ ∂²/∂φ² + 1/r²sinθ ∂/∂θsinθ ∂/∂θ]u = V (r, θ, φ)u, 

where 

V(r, θ, φ) = V₁(r) + V₂(φ)/r²sin²θ + V₃(θ)/r².

Perform separation of variables and reduce the equation into three ordinary differential equations.

Answer 1

Substituting 

u = R(r)Θ(θ)Φ(φ) 

into the PDE and multiplying the resulting equation by r², we get