How to force Mathematica to evaluate a limit in the boundary conditions of a differential equation?

Question

I'm experiencing troubles in obtaining the result of an ordinary differential equation with a boundary condition at infinity. I write down

Limit[DSolve[{ψ''[r] + 2/r*ψ'[r] - κ^2*ψ[r] == 
    0, ψ[a/2] == Subscript[Psi, 0], ψ[M] == 0}, ψ[r], 
  r], M -> Infinity]

And obtain

\!\(\*UnderscriptBox[\(\[Limit]\), \(M \[Rule] ∞\)]\) {{\
ψ[r] -> (
    a E^((a κ)/2 - 
      r κ) (-E^(2 M κ) + E^(2 r κ)) Subscript[
     Psi, 0])/(2 (E^(a κ) - E^(2 M κ)) r)}}

Which is correct, but is not evaluated for the limit. How can I force Mathematica to evaluate this limit?

Answer 1

Trying to do to much at once.

ψ[r_] = ψ[r] /. 
  DSolve[{(ψ'')[r] + (2 Derivative[1][ψ][r])/ r - κ^2 ψ[r] == 0, ψ[a/2] == 
      Psi0, ψ[M] == 0}, ψ[r], r][[1]]

Limit[ψ[r], M -> ∞]

Simplify[%, {r > 0 && κ > 0 && a > 0 && Psi0 ∈ Reals}]
(*(a Psi0 E^(1/2 κ (a - 2 r)))/(2 r)*)

Subscripts are not a good idea and Mathematica cannot always perform every function on rules.