Hi, It is true that in general these problems are tough. There are Maple commands to treat them though. I mean: "Given a system of equations, differential or not, linear or not, involving inequations or not, depending on some parameters, say {a,b,c,...}, such that a solution exists only for some particular values of these parameters, compute all these solutions as well as the different values of the parameters {a,b,c,...} such that these solutions exist". By the way, among the existing computer algebra systems, Maple is the only one able to handle such a general problem. The Maple packages handling these probelms are diffalg and RIF. They can do this type of computation since Maple R5 (diffalg) and Maple 6 (RIF), and both work rather efficiently, in my opinion RIF performs better on average problems. The computation using diffalg/RIF in Maple is also simplified and extended, by means of the PDEtools[casesplit] and PDEtools[dpolyform] commands, in order to handle basically all the mathematical functions. That is, to perform types of (maybe differential) Grobner basis elimination on systems not just rational in the unknowns and the independent variables. Despite being in PDEtools, casesplit handles the same way systems of equations not involving derivatives. To see some examples of how this works see ?PDEtools[casesplit] and PDEtools[dpolyform]. Edgardo S. Cheb-Terrab Research Fellow, Maplesoft Editor for Computer Algebra, Computer Physics Communications

Hi, It is true that in general these problems are tough. There are Maple commands to treat them though. I mean: "Given a system of equations, differential or not, linear or not, involving inequations or not, depending on some parameters, say {a,b,c,...}, such that a solution exists only for some particular values of these parameters, compute all these solutions as well as the different values of the parameters {a,b,c,...} such that these solutions exist". By the way, among the existing computer algebra systems, Maple is the only one able to handle such a general problem. The Maple packages handling these probelms are diffalg and RIF. They can do this type of computation since Maple R5 (diffalg) and Maple 6 (RIF), and both work rather efficiently, in my opinion RIF performs better on average problems. The computation using diffalg/RIF in Maple is also simplified and extended, by means of the PDEtools[casesplit] and PDEtools[dpolyform] commands, in order to handle basically all the mathematical functions. That is, to perform types of (maybe differential) Grobner basis elimination on systems not just rational in the unknowns and the independent variables. Despite being in PDEtools, casesplit handles the same way systems of equations not involving derivatives. To see some examples of how this works see ?PDEtools[casesplit] and PDEtools[dpolyform]. Edgardo S. Cheb-Terrab Research Fellow, Maplesoft Editor for Computer Algebra, Computer Physics Communications

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