differential equations - DSolve will not apply assumption m ∈ Integers

I am trying to solve a linear second order ODE using DSolve which involves an arbitrary integer m. DSolve gives me a solution when I set m to a particular integer (I have tried several, including negative/positive, even/odd, and 0). When I try to use the assumption m ∈ Integers and ask DSolve to solve this ODE for an arbitrary integer m, it does not work. Here is the input:

$Assumptions = m ∈ Integers
testk = 
  0 == -16 c m^2 Cos[x] k[x] - c (-7 Sin[x] + Sin[3x]) k'[x] 
       + Cos[x] Sin[x]^2 (m^2 (3 + 4 m Cos[x] + Cos[2 x]) Tan[x/2]^(2 m) + 4 c k''[x])

DSolve[ testk, k[x], x]

DSolve[ 0 == -16 c m^2 Cos[x] k[x] - c (-7 Sin[x] + Sin[3 x]) k'[x] 
              + Cos[x] Sin[x]^2 (m^2 (3 + 4 m Cos[x] + Cos[2 x]) Tan[x/2]^(2 m)
              + 4 c k''[x], k[x], x]

Update: I have realized that the problem is that DSolve does not apply any of the global assumptions. Does anyone know how I can make DSolve apply the assumption that m ∈ Integers? I have tried using Assuming[,] to set the assumptions locally but that did not work either.