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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.

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1 Answer 1

Not sure if this will help, but you can transform your ODE to have rational coefficients by subbing $t = \cos x$, which gives

$k'(x)=-\sqrt{1-t^2} k'(t), \quad k''(x)=(1-t^2)k''(t)-t k'(t), \;\; \text{ and } \;\; x=\cos ^{-1}(t)$:

testk=-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]);

$fromTrig={k'[x]:>(-Sqrt[1-t^2])*k'[t],k[x]->k[t],k''[x]->-t*k'[t]+k''[t]-t^2*k''[t],x->ArcCos[t]};

simped = Simplify[FunctionExpand[testk/.$fromTrig],-1<t<1&&m\[Element]Integers];

Collect[(1+t)^m PowerExpand[simped], {k[t],k'[t],k''[t]}]==0
-2 m^2 (1-t)^m t (-1+t^2) (1+2 m t+t^2)-16 c m^2 t (1+t)^m k[t] + 
  8 c (1+t)^m (-1+t^2) k'[t]+4 c t (1+t)^m (-1+t^2)^2 k''[t] == 0
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