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Re: Solving a non-linear system to find coefficients

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113035] Re: Solving a non-linear system to find coefficients
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Tue, 12 Oct 2010 04:23:04 -0400 (EDT)

Don't enter the c[2] and a[3,1] using subscripts.

Needs["NumericalDifferentialEquationAnalysis`"]

eqs = RungeKuttaOrderConditions[4, 4, ButcherRowSum -> True, 
   RungeKuttaMethod -> Explicit];

Solve[Flatten[Join[eqs, {c[2] == 1/2, a[3, 1] == 0}]]]


Bob Hanlon

---- Allamarein <matteo.diplomacy at gmail.com> wrote: 

=============
Like suggested, I have just tried:

Needs["NumericalDifferentialEquationAnalysis`"]

eqs = RungeKuttaOrderConditions[4, 4, ButcherRowSum -> True,
   RungeKuttaMethod -> Explicit];

Solve[Flatten[
  Join[eqs, {Subscript[c, 2] == 1/2, Subscript[a, 3, 1] == 0}]]]

My "adviser"  reports  I should get this output (as a matter of fact
it is the correct result):
{{Subscript[b, 1] -> 1/6, Subscript[a, 2, 1] -> 1/2,
  Subscript[a, 4, 1] -> 0, Subscript[a, 3, 1] -> 0,
  Subscript[b, 2] -> 1/3, Subscript[a, 4, 2] -> 0,
  Subscript[a, 3, 2] -> 1/2, Subscript[a, 4, 3] -> 1,
  Subscript[b, 3] -> 1/3, Subscript[c, 4] -> 1,
  Subscript[b, 4] -> 1/6, Subscript[c, 3] -> 1/2,
  Subscript[c, 2] -> 1/2}}

Instead I get this puzzled result:
{{Subscript[b, 1] -> 1/6, Subscript[a, 2, 1] -> 1/2,
  Subscript[a, 3, 1] -> (-1 + 3 Subscript[b, 3])/(6 Subscript[b, 3]),
  Subscript[a, 4, 1] -> 0, Subscript[c, 2] -> 1/2,
  Subscript[a, 3, 1] -> 0,
  Subscript[b, 2] -> 1/3 (2 - 3 Subscript[b, 3]),
  Subscript[a, 4, 2] -> 1 - 3 Subscript[b, 3],
  Subscript[a, 3, 2] -> 1/(6 Subscript[b, 3]),
  Subscript[a, 4, 3] -> 3 Subscript[b, 3], Subscript[c, 4] -> 1,
  Subscript[b, 4] -> 1/6, Subscript[c, 3] -> 1/2,
  Subscript[c, 2] -> 1/2}, {Subscript[b, 1] -> 1/6,
  Subscript[a, 2, 1] -> 1, Subscript[a, 3, 1] -> 3/8,
  Subscript[a, 4, 1] -> (-1 + 4 Subscript[b, 4])/(4 Subscript[b, 4]),
  Subscript[c, 2] -> 1/2, Subscript[a, 3, 1] -> 0,
  Subscript[b, 2] -> 1/6 (1 - 6 Subscript[b, 4]),
  Subscript[a, 4, 2] -> -(1/(12 Subscript[b, 4])),
  Subscript[a, 3, 2] -> 1/8,
  Subscript[a, 4, 3] -> 1/(3 Subscript[b, 4]), Subscript[b, 3] -> 2/3,
   Subscript[c, 4] -> 1, Subscript[c, 3] -> 1/2,
  Subscript[c, 2] -> 1}, {Subscript[b, 1] ->
   1/6 (1 - 6 Subscript[b, 3]), Subscript[a, 2, 1] -> 1/2,
  Subscript[a, 3, 1] -> -(1/(12 Subscript[b, 3])),
  Subscript[a, 4, 1] -> 1/2 (-1 - 12 Subscript[b, 3]),
  Subscript[c, 2] -> 1/2, Subscript[a, 3, 1] -> 0,
  Subscript[b, 2] -> 2/3, Subscript[a, 4, 2] -> 3/2,
  Subscript[a, 3, 2] -> 1/(12 Subscript[b, 3]),
  Subscript[a, 4, 3] -> 6 Subscript[b, 3], Subscript[c, 4] -> 1,
  Subscript[b, 4] -> 1/6, Subscript[c, 3] -> 0,
  Subscript[c, 2] -> 1/2}, {Subscript[b, 1] ->
   1/12 (6 - (4 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) + (
      2 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) + 1/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (2 Subscript[c, 2])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (2 Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (
      8 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c,
       3])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      4 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c, 3])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]))),
  Subscript[a, 2, 1] -> Subscript[c, 2],
  Subscript[a, 3, 1] ->
   Subscript[c,
    3] - ((Subscript[c, 2] - Subscript[c, 3]) Subscript[c, 3])/(
    Subscript[c, 2] (-2 + 4 Subscript[c, 2])),
  Subscript[a, 4, 1] ->
   1/12 (-38 + 60 Subscript[c, 2] - (
      16 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) + (
      8 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) + 6/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (6 Subscript[c, 2])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (6 Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (
      70 Subscript[c, 2] Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (60 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (
      40 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c,
       3])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      20 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c, 3])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) -
      3/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      3 Subscript[c,
       2])/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      46 (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((3 -
         4 Subscript[c, 2] - 4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      32 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c,
       3] (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/(
      Subscript[c,
       2] (-2 + 4 Subscript[c, 2]) (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (180 \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)] (-1 + 3 Subscript[c, 2] - 2
\!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((3 -
         4 Subscript[c, 2] - 4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      3 (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3])^2 (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      3 Subscript[c, 2] (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3])^2 (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      52 (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      46 Subscript[c, 2] (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]))), Subscript[c, 2] -> 1/2,
  Subscript[a, 3, 1] -> 0,
  Subscript[b, 2] ->
   1/12 (1/(
      1 - Subscript[c, 2] - Subscript[c, 3] +
       Subscript[c, 2] Subscript[c, 3]) - (2 Subscript[c, 3])/(
      1 - Subscript[c, 2] - Subscript[c, 3] +
       Subscript[c, 2] Subscript[c, 3]) + (
      4 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) - (
      2 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) + 1/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (
      8 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c,
       3])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      4 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c, 3])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) -
      1/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + Subscript[c,
      2]/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]))),
  Subscript[a, 4, 2] ->
   1/12 (50 - 60 Subscript[c, 2] + (
      16 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) - (
      8 (Subscript[c, 2] - Subscript[c, 3]) \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])^2) - 6/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (6 Subscript[c, 2])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (6 Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (
      70 Subscript[c, 2] Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (60 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (
      40 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c,
       3])/((-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      20 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c, 3])/(
      Subscript[c,
       2] (-2 +
         4 Subscript[c, 2]) (Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) +
      3/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      3 Subscript[c,
       2])/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      58 (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((3 -
         4 Subscript[c, 2] - 4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      32 (Subscript[c, 2] - Subscript[c, 3]) Subscript[c,
       3] (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/(
      Subscript[c,
       2] (-2 + 4 Subscript[c, 2]) (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (180 \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)] (-1 + 3 Subscript[c, 2] - 2
\!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((3 -
         4 Subscript[c, 2] - 4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      3 (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3])^2 (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      3 Subscript[c, 2] (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3])^2 (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) + (
      52 (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - (
      46 Subscript[c, 2] (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
         3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c, 3]))/((1 -
         Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (3 - 4 Subscript[c, 2] -
         4 Subscript[c, 3] +
         6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2]
           Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]))),
  Subscript[a, 3,
   2] -> ((Subscript[c, 2] - Subscript[c, 3]) Subscript[c, 3])/(
   Subscript[c, 2] (-2 + 4 Subscript[c, 2])),
  Subscript[a, 4, 3] -> (-1 + 3 Subscript[c, 2] - 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] + Subscript[c, 3] -
    3 Subscript[c, 2] Subscript[c, 3] + 2 \!
\*SubsuperscriptBox[\(c\), \(2\), \(2\)] Subscript[c,
     3])/((3 - 4 Subscript[c, 2] - 4 Subscript[c, 3] +
      6 Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[c,
        3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])),
  Subscript[b, 3] ->
   1/12 (2/(
      1 - Subscript[c, 2] - Subscript[c, 3] +
       Subscript[c, 2] Subscript[c, 3]) - (2 Subscript[c, 2])/(
      1 - Subscript[c, 2] - Subscript[c, 3] +
       Subscript[c, 2] Subscript[c, 3]) - 2/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) + (2 Subscript[c, 2])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) - (2 Subscript[c, 3])/(
      Subscript[c, 2] Subscript[c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]) +
      1/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)])) - Subscript[c,
      2]/((1 - Subscript[c, 2] - Subscript[c, 3] +
         Subscript[c, 2] Subscript[c, 3]) (Subscript[c, 2] Subscript[
          c, 3] - \!
\*SubsuperscriptBox[\(c\), \(3\), \(2\)]))), Subscript[c, 4] -> 1,
  Subscript[b, 4] -> (
   3 - 4 Subscript[c, 2] - 4 Subscript[c, 3] +
    6 Subscript[c, 2] Subscript[c, 3])/(
   12 (-1 + Subscript[c, 2]) (-1 + Subscript[c, 3]))}}




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