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Solving system of equations gives no solution

Zaza Esta

Posted 6 months ago

I am trying to solve the equations below. But Mathematica doesn't give any solutions. Anyone can help about. Thank you. eqs= {(b[0] + b[2]) (b[3] + b[5]) + d[0] == 1, (b[3] + b[5]) (b[1] + b[2] c[1]) + (b[0] + b[2]) (b[4] + b[5] c[1]) + d[1] == 1, b[1] (b[4] + b[5] c[1]) + c[1] (b[0] b[5] c[1] + b[2] (b[4] + (b[3] + 3 b[5]) c[1])) == 1/2, c[1]^2 (b[5] (b[1] + b[0] c[1]) + b[2] (b[4] + (b[3] + 4 b[5]) c[1])) == 1/6, c[1]^3 (b[5] (b[1] + b[0] c[1]) + b[2] (b[4] + (b[3] + 5 b[5]) c[1])) == 1/24, c[1]^4 (b[5] (b[1] + b[0] c[1]) + b[2] (b[4] + (b[3] + 6 b[5]) c[1])) == 1/120, c[1]^5 (b[5] (b[1] + b[0] c[1]) + b[2] (b[4] + (b[3] + 7 b[5]) c[1])) == 1/720, c[1]^6 (b[5] (b[1] + b[0] c[1]) + b[2] (b[4] + (b[3] + 8 b[5]) c[1])) == 1/5040, c[1]^7 (b[5] (b[1] + b[0] c[1]) + b[2] (b[4] + (b[3] + 9 b[5]) c[1])) == 1/40320} sol = Solve[ eq1, {b[5], b[4], b[3], b[2], b[1], b[0], d[0], d[1], c[1]}]

I am trying to solve the equations below. But Mathematica doesn't give any solutions. Anyone can help about. Thank you.

eqs= {(b[0] + b[2]) (b[3] + b[5]) + d[0] == 
  1, (b[3] + b[5]) (b[1] + b[2] c[1]) + (b[0] + b[2]) (b[4] + 
      b[5] c[1]) + d[1] == 1, 
 b[1] (b[4] + b[5] c[1]) + 
   c[1] (b[0] b[5] c[1] + b[2] (b[4] + (b[3] + 3 b[5]) c[1])) == 1/2, 
 c[1]^2 (b[5] (b[1] + b[0] c[1]) + 
     b[2] (b[4] + (b[3] + 4 b[5]) c[1])) == 1/6, 
 c[1]^3 (b[5] (b[1] + b[0] c[1]) + 
     b[2] (b[4] + (b[3] + 5 b[5]) c[1])) == 1/24, 
 c[1]^4 (b[5] (b[1] + b[0] c[1]) + 
     b[2] (b[4] + (b[3] + 6 b[5]) c[1])) == 1/120, 
 c[1]^5 (b[5] (b[1] + b[0] c[1]) + 
     b[2] (b[4] + (b[3] + 7 b[5]) c[1])) == 1/720, 
 c[1]^6 (b[5] (b[1] + b[0] c[1]) + 
     b[2] (b[4] + (b[3] + 8 b[5]) c[1])) == 1/5040, 
 c[1]^7 (b[5] (b[1] + b[0] c[1]) + 
     b[2] (b[4] + (b[3] + 9 b[5]) c[1])) == 1/40320}
sol = Solve[
  eq1, {b[5], b[4], b[3], b[2], b[1], b[0], d[0], d[1], c[1]}]

POSTED BY: Zaza Esta

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 6 months ago

Your system has no solution. You can prove it with these combinations of equations # 4,5,6,7: e1 = AddSides[ AddSides[ MultiplySides[eqs[[5]], 3 c[1]^2, GenerateConditions -> False], MultiplySides[eqs[[4]], -2 c[1]^3, GenerateConditions -> False]], MultiplySides[eqs[[7]], -1]] // Expand e2 = AddSides[ AddSides[ MultiplySides[eqs[[6]], 3/2 c[1], GenerateConditions -> False], MultiplySides[eqs[[4]], -(1/2) c[1]^3, GenerateConditions -> False]], MultiplySides[eqs[[7]], -1]] // Expand

Your system has no solution. You can prove it with these combinations of equations # 4,5,6,7:

e1 = AddSides[
   AddSides[
    MultiplySides[eqs[[5]], 3 c[1]^2, GenerateConditions -> False], 
    MultiplySides[eqs[[4]], -2 c[1]^3, GenerateConditions -> False]],
   MultiplySides[eqs[[7]], -1]] // Expand
e2 = AddSides[
   AddSides[
    MultiplySides[eqs[[6]], 3/2 c[1], GenerateConditions -> False], 
    MultiplySides[eqs[[4]], -(1/2) c[1]^3, 
     GenerateConditions -> False]],
   MultiplySides[eqs[[7]], -1]] // Expand

POSTED BY: Gianluca Gorni

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 6 months ago

It seems that the equations number 4,5, and 6 are not compatible with the next ones: sol456 = Solve[eqs[[{4, 5, 6}]]] Simplify[eqs[[7]] /. sol456] Simplify[eqs[[8]] /. sol456] Simplify[eqs[[9]] /. sol456]