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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