calculus and analysis - Working with a system of differential equations that cannot be solved explicitly

I have to work a lot with three functions $\;o_1(t), o_2(t), o_3(t)\;$ that are solutions to a certain system of differential equations:

Halp = {
  D[o1[t], t] == o1[t]*o2[t] + o1[t]*o3[t] - o2[t]*o3[t], 
  D[o2[t], t] == o1[t]*o2[t] + o2[t]*o3[t] - o1[t]*o3[t],
  D[o3[t], t] == o1[t]*o3[t] + o2[t]*o3[t] - o1[t]*o2[t]}
sol = DSolve[Halp, {o1[t], o2[t], o3[t]}, {t}]

The system cannot be solved explicitly with Mathematica, but I do not need the solution. What I would like to do is to compute higher order derivatives assuming that $o_1(t)$, $o_2(t)$, and $o_3(t)$ are solutions to the above system.

How could I tell Mathematica to do this?

Answer

There are many ways which one could exploit. For example let's use calculate o1''[t] eliminating o1'[t], o2'[t], o3'[t]. We can use Eliminate:

#1/.( Eliminate[ Join[ D[Halp, t], Halp], {#2, #3}] //ToRules)&[o1''[t], o2''[t], o3''[t]]

2 o1[t] o2[t]^2 + 2 o1[t] o2[t] o3[t] - 2 o2[t]^2 o3[t] + 2 o1[t] o3[t]^2 - 2 o2[t] o3[t]^2 

Changing the order in the square bracket we can calculate o2''[t] and o3''[t].

For the third order derivative of e.g. o1[t] we can eliminate other dependent variables in many different ways, let's point out one of them:

#1 /.( 
Eliminate[ Join[ D[Halp, {t, 2}], D[Halp, t], Halp], {##2}] 
//ToRules) &[o1'''[t], o2'''[t], o3'''[t], o2''[t], o3''[t], o1''[t]]

  2 o1[t]^2 o2[t]^2 + 4 o1[t] o2[t]^3 - 4 o1[t]^2 o2[t] o3[t] + 8 o1[t] o2[t]^2 o3[t]
- 4 o2[t]^3 o3[t] + 2 o1[t]^2 o3[t]^2 + 8 o1[t] o2[t] o3[t]^2 - 10 o2[t]^2 o3[t]^2 
+ 4 o1[t] o3[t]^3 - 4 o2[t] o3[t]^3  

In the above we eliminated all dependent variables starting from the second position in the square bracket (note very useful sign ##n i.e. SlotSequence ) and calculated o1'''[t]. Of course you can try to get rid of different dependent variables. In general this is a difficult issue to decide which dependent variables could be eliminated and how they could be represented by another variables. I suggest to take a closer look at GroebnerBasis with the MonomialOrder -> EliminationOrder option.

GroebnerBasis[ polys, vars, elims, MonomialOrder -> EliminationOrder]

If given equations are relatively simple you can't observe a common problem in differential elimination - so called intermediate expresssion swell. For interesting mathematical issues related to the problem I'd suggest to study e.g. a recent monograph Involution by Werner Seiler (Springer 2009). However I couldn't mention interesting packages in Mathematica related to differential elimination (there are such packages e.g. in Maple).