Asymptotics and limits of second order ODE which depend on (two) parameters

Question

Dear Mathematica community, for this second order ODE:

h''[x] Sinh[2 x] + h'[x] 2 Cosh[2 x] - 2 h[x] Tanh[x] == 0, 

which is basically the harmonic equation for the radial part in certain non-Euclidean metric, I need

Limit[h'[x] Sinh[2 x]/2, x -> Infinity]

or even better, express the asymptotic of h[x] and h'[x] in terms of exponential functions as x -> Infinity. I only need the largest term. In my home edition this asymptotic of h'[x] as x-> Infinity exceeds Recursion depth and Mathematica stops, unfortunately. Plug in the initial condition h'[0]=0 also stops Mathematica. For the above limit I get 0. Now generally the ODE depends on 2 parameters

h''[x] Sinh[2 x] + h'[x] 2 Cosh[2 x] - 2 m^2 h[x] Tanh[x]/a^2 == 0

Here m is a natural number, a is a small positive number. I want to find an upper of

Limit[a^2 h'[x] Sinh[2 x]/2, x -> Infinity]

which I hope can convince me that as a->0, the limit is 0 (or the limit is 0 regardless of a^2. Using approximation, like replacing Sinh[x] by Exp[x], I get a solution whose asymptotic computation does not exceed recursion limit:

{{TerminatedEvaluation[
"RecursionLimit"] -> -((
 Sqrt[E^(-2 x)] BesselI[1, Sqrt[2] Sqrt[E^(-2 x)]] C[1])/Sqrt[
 2]) + Sqrt[2] Sqrt[E^(-2 x)]
  BesselK[1, Sqrt[2] Sqrt[E^(-2 x)]] C[2]}}

But still my Mathematica cannot handle it when it has extra parameters a or m. More importantly, theoretically I am not sure at this moment approximating ODE with another ODE gives me a solution with the same asymptotic. Thank you!

Answer 1

You could find the solution $h(x)$ and then use Asymptotic on its derivative?

ode = h''[x] Sinh[2 x] + h'[x] 2 Cosh[2 x] - 2 h[x] Tanh[x] == 0
sol = h[x] /. First@DSolve[ode, h[x], x]
res = D[sol, x]*Sinh[2*x]/2

Asymptotic[res, x -> Infinity] // Simplify