Hi, You question is about aspects of a numerical solution, assuming that

> ... for this problem no closed-form...

So maybe this is of help: in Maple 9.5 and 10 (perhaps also in previous versions), dsolve can compute a closed-form solution for this problem: > dsolve([diff(s(t),t) = A - A * rho^((1 - r^(theta * t)) * x) - v, diff(f(t),t)=((c - s(t)) / l) * m + x, diff(h(t),t) = x, f(0) = 0, s(0) = 0, h(0) = 0]);

                           x          (theta t)
                      A rho  Ei(1, x r          ln(rho))
  {h(t) = x t, s(t) = ----------------------------------
                                 theta ln(r)

                                                          t
                x                                        /
           A rho  Ei(1, x ln(rho))                      |   m c
         - ----------------------- + (A - v) t, f(t) =  |   ---
                 theta ln(r)                            |    l
                                                       /
                                                         0

                            x          (theta _z1)
           m A _z1   m A rho  Ei(1, x r            ln(rho))   m v _z1
         - ------- - -------------------------------------- + -------
              l                  l theta ln(r)                   l

                  x
           m A rho  Ei(1, x ln(rho))
         + ------------------------- + x d_z1}
                 l theta ln(r)

Edgardo S. Cheb-Terrab Research Fellow, Maplesoft