Find the constant which include with equation after

Question

I want to solve an equation. I want to get all steps for the solution with details and explnation.

  Ψ(r, y, t) = (Rr)^1/2/2ΠK{(2-k²) K(k)-2E(k)} - (Rr)^1/2/2ΠK {(2 - k²) K(k^-) - 2E(k^-)},

where

k² = 4Rr/(y - Y)² + (r + R)², k^-2 = 4Rr/(y + Y)² + (r + R)^2'

Where K and E are complete elliptic integrals of the first and second kind.

K(k) = ₀∫^1/2Π(1 - k² sin² dx)^-1/2dx, E(k) = ₀∫^1/2Π(1 - k² sin²x)^1/2 dx.

Ur = (1/r * ∂Ψ/∂y)

Uy = (-1/r * ∂Ψ/∂y)

∂Φ/∂r = Ur

∂Φ/∂y = Uy

∂Φ/∂y = Uy = 0 at y = 0

I want to find (Φ). After that find the constant which include with equation after integral.

Differentiated Φ with respect to y and apply

∂Φ/∂y = Uy = 0 at y = 0