DSolve not outputting any result

Question

I'm trying to solve a system of differential equations using DSolve, but I am not getting any output beyond a repetition of my input:

DSolve[
 {p''[t] + 2 Cosh[c[t]] c'[t] p'[t]/Sinh[c[t]] == 0, 
  c''[t] - Sinh[c[t]] Cosh[c[t]] (p'[t])^2 == 0,
  c[0] == b, p[0] == 0,
  c'[0] == 0, p'[0] == 1},
 {c[t], p[t]}, t]

I saw this question already, but I don't have any floating point numbers so that can't be the issue here. Why is DSolve not returning anything here?

Answer 1

This set of equations certainly is solvable analytically, although DSolve seems unable to do so without assistance. Begin by solving the first equation only, applying the relevant boundary conditions, and solving algebraically for p'[t].

Solve[Quiet@DSolve[{p''[t] + 2 Cosh[c[t]] c'[t] p'[t]/Sinh[c[t]] == 0, 
       c[0] == b}, c[t], t][[1, 1]] /. {p'[0] -> 1, Rule -> Equal}, p'[t]][[1, 1]]
(* Derivative[1][p][t] -> Csch[c[t]]^2 Sinh[b]^2 *)

This expression can be substituted into the second equation to yield a second order ODE for c[t] only, and this can be solved with DSolve, although applying the remaining boundary conditions is not straightforward. So, consider multiplying the new ODE by c'[t] and integrating just once.

C[1] == Integrate[(c''[t] - Sinh[c[t]] Cosh[c[t]] (p'[t])^2 /. %) c'[t], t]
(* C[1] == 1/2 Coth[c[t]]^2 Sinh[b]^4 + 1/2 Derivative[1][c][t]^2 *)

The constant then can be evaluated to yield a first order ODE,

% /. t -> 0 /. {c[0] -> b, c'[0] -> 0}
%% /. (% /. Equal -> Rule)
(* 1/2 Cosh[b]^2 Sinh[b]^2 == 1/2 Coth[c[t]]^2 Sinh[b]^4 + 1/2 Derivative[1][c][t]^2 *)

which I must leave to the OP to solve, since I am out of time.