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Engineering Mathematics

How to solve this one?

Jie Li

Posted 9 years ago

c23 == c13 + a13, c22 == c12 + a12, c21 == c11 + a11, b23 == 2 Aa13, b22 == 2 Aa12, b21 == 2 A a11 , b33 == B(2 a23 + b23), b32 == B(2 a22 + b22), b31 == B (2a21 + b21 ), c33 == a23 + b23 + c23, c32 == a22 + b22 + c22, c31 == a21 + b21 + c21 , a33 == -0.5b33, a32 == -0.5b32, a31 == -0.5 b31 t, a22/3 + b22/2 + c22 + Fb23 == E(2 a23 + b23), (2 a21 )/3 + b21 + 2 c21 + F b22 == E(2 a22 + b22), F b21 t == E (2 a21 t + b21 t), a32/3 + b32/2 + c32 + H == G b33, (2 a31 )/3 + b31 + 2 c31 == G b32, 0 == b31 , a12/3 + c12 == D + 2 C a13, 2 c11 + (a112)/3 == 2 C *a12, 0 == a11 how to calculate a11 to c33? (there is no b11, b12,b13)

POSTED BY: Jie Li

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Jim Baldwin

Jim Baldwin, Retired

Posted 9 years ago

Not sure what how you're getting the above. Are you typing in the commands rather than copying-and-pasting? It is "Solve" rather than "solve", "gb32" should be "g b32", "gb21" should be "g b21", "a312/3" should be "a31 2/3", "2ea23" should be "2 e a23". I suggest using the Code Sample input or better yet attach your working notebook. But if the changes I made are the correct ones, then there is no unique solution for all parameters. You can choose anything you want for c23. In[5]:= Solve[{c23 == c13 + a13, c22 == c12 + a12, c21 == c11 + a11, c33 == a23 + b23 + c23, c32 == a22 + b22 + c22, c31 == a21 + b21 + c21, b23 == 2 a a13, b22 == 2 a a12, b21 == 2 a a11, b33 == b (2 a23 + b23), b32 == b (2 a22 + b22), b31 == b (2 a21 + b21), a33 == -b33/2, a32 == -b32/2, a31 == -b31 t/2, a11 == 0, 2 c a12 - a11 2/3 - 2 c11 == 0, d - a12/3 + c12 == 0, 2 e a21 + (e - f) b21 == 0, 2 e a22 + (e - f) b22 - a21 2/3 - b21 - 2 c21 == 0, 2 e a23 + (e - f) b23 - a22/3 - b22/2 - c22 == 0, g b21 == 0, g b32 - a31 2/3 - b31 - 2 c31 == 0, g*b33 - h - a32/3 - b32/2 - c32 == 0}, {a11, a12, a13, a21, a22, a23, a31, a32, a33, b21, b22, b23, b31, b32, b33, c11, c12, c13, c21, c22, c23, c31, c32, c33}] During evaluation of In[5]:= Solve::svars: Equations may not give solutions for all "solve" variables. >> Out[5]= {{a11 -> 0, a12 -> 0, a13 -> -((d e - b d g - e h)/(2 a b f g)), a21 -> 0, a22 -> 0, a23 -> -((-d e + d f + b d g + e h - f h)/(2 b f g)), a31 -> 0, a32 -> 0, a33 -> -((-d + h)/(2 g)), b21 -> 0, b22 -> 0, b23 -> -((d e - b d g - e h)/(b f g)), b31 -> 0, b32 -> 0, b33 -> -((d - h)/g), c11 -> 0, c12 -> -d, c13 -> c23 - (-d e + b d g + e h)/(2 a b f g), c21 -> 0, c22 -> -d, c31 -> 0, c32 -> -d, c33 -> c23 - (d e + d f - b d g - e h - f h)/(2 b f g)}}

Not sure what how you're getting the above. Are you typing in the commands rather than copying-and-pasting? It is "Solve" rather than "solve", "gb32" should be "g b32", "gb21" should be "g b21", "a312/3" should be "a31 2/3", "2ea23" should be "2 e a23". I suggest using the Code Sample input or better yet attach your working notebook.

But if the changes I made are the correct ones, then there is no unique solution for all parameters. You can choose anything you want for c23.

In[5]:= Solve[{c23 == c13 + a13,
  c22 == c12 + a12,
  c21 == c11 + a11,
  c33 == a23 + b23 + c23,
  c32 == a22 + b22 + c22,
  c31 == a21 + b21 + c21,
  b23 == 2 a a13,
  b22 == 2 a a12,
  b21 == 2 a a11,
  b33 == b (2 a23 + b23),
  b32 == b (2 a22 + b22),
  b31 == b (2 a21 + b21),
  a33 == -b33/2,
  a32 == -b32/2,
  a31 == -b31 t/2,
  a11 == 0,
  2 c a12 - a11 2/3 - 2 c11 == 0,
  d - a12/3 + c12 == 0,
  2 e a21 + (e - f) b21 == 0,
  2 e a22 + (e - f) b22 - a21 2/3 - b21 - 2 c21 == 0, 
  2 e a23 + (e - f) b23 - a22/3 - b22/2 - c22 == 0,
  g b21 == 0,
  g b32 - a31 2/3 - b31 - 2 c31 == 0,
  g*b33 - h - a32/3 - b32/2 - c32 == 0}, {a11, a12, a13, a21, a22, 
  a23, a31, a32, a33, b21, b22, b23, b31, b32, b33, c11, c12, c13, 
  c21, c22, c23, c31, c32, c33}]


During evaluation of In[5]:= Solve::svars: Equations may not give solutions for all "solve" variables. >>

Out[5]= {{a11 -> 0, a12 -> 0, 
  a13 -> -((d e - b d g - e h)/(2 a b f g)), a21 -> 0, a22 -> 0, 
  a23 -> -((-d e + d f + b d g + e h - f h)/(2 b f g)), a31 -> 0, 
  a32 -> 0, a33 -> -((-d + h)/(2 g)), b21 -> 0, b22 -> 0, 
  b23 -> -((d e - b d g - e h)/(b f g)), b31 -> 0, b32 -> 0, 
  b33 -> -((d - h)/g), c11 -> 0, c12 -> -d, 
  c13 -> c23 - (-d e + b d g + e h)/(2 a b f g), c21 -> 0, c22 -> -d, 
  c31 -> 0, c32 -> -d, 
  c33 -> c23 - (d e + d f - b d g - e h - f h)/(2 b f g)}}

POSTED BY: Jim Baldwin

Jim Baldwin

Jim Baldwin, Retired

Posted 9 years ago

Using the "Code sample" input (icon on the upper left just below "Reply to this discussion") will make the code easier to read. I made some changes in what you wrote to match what Mathematica requires so I might very well have made wrong changes. There needs to be spaces between some of the entries such as Fb23 needs to be "F b23" or "F * b23". I also made all of the capital letters into lowercase as sometimes uppercase might clash with an existing Mathematica function. And example of this is "E" which gets you 2.71828. And finally there is "(a112)/3" which I assume should either be "a11/3" or "a11 * 2/3". (I used "a11*2/3" in the example below.) In[8]:= Solve[{c23 == c13 + a13, c22 == c12 + a12, c21 == c11 + a11, b23 == 2 a a13, b22 == 2 a a12, b21 == 2 a a11, b33 == b (2 a23 + b23), b32 == b (2 a22 + b22), b31 == b (2 a21 + b21), c33 == a23 + b23 + c23, c32 == a22 + b22 + c22, c31 == a21 + b21 + c21, a33 == -b33/2, a32 == -b32/2, a31 == -b31 t/2, a22/3 + b22/2 + c22 + f b23 == e (2 a23 + b23), (2 a21)/3 + b21 + 2 c21 + f b22 == e (2 a22 + b22), f b21 t == e (2 a21 t + b21 t), a32/3 + b32/2 + c32 + h == g b33, (2 a31)/3 + b31 + 2 c31 == g b32, 0 == b31, a12/3 + c12 == d + 2 c a13, 2 c11 + a11 2/3 == 2 c a12, 0 == a11}, {a11, a12, a13, a21, a22, a23, a31, a32, a33, b21, b22, b23, b31, b32, b33, c11, c12, c13, c21, c22, c23, c31, c32, c33}] During evaluation of In[8]:= Solve::svars: Equations may not give solutions for all "solve" variables. >> Out[8]= {{a11 -> 0, a12 -> 0, a13 -> -((d e - b d g + e h)/(2 (c e - b c g - a b f g))), a21 -> 0, a22 -> 0, a23 -> -((-a d e + a d f + a b d g + c h - a e h + a f h)/( 2 (c e - b c g - a b f g))), a31 -> 0, a32 -> 0, a33 -> (b (a d f + c h + a f h))/(2 (c e - b c g - a b f g)), b21 -> 0, b22 -> 0, b23 -> -((a d e - a b d g + a e h)/(c e - b c g - a b f g)), b31 -> 0, b32 -> 0, b33 -> (b (a d f + c h + a f h))/(-c e + b c g + a b f g), c11 -> 0, c12 -> -((a b d f g + c e h)/(c e - b c g - a b f g)), c13 -> c23 - (-d e + b d g - e h)/(2 (c e - b c g - a b f g)), c21 -> 0, c22 -> -((a b d f g + c e h)/(c e - b c g - a b f g)), c31 -> 0, c32 -> -((a b d f g + c e h)/(c e - b c g - a b f g)), c33 -> c23 - (a d e + a d f - a b d g + c h + a e h + a f h)/( 2 (c e - b c g - a b f g))}} So either I've incorrectly modified what you had or there is not a single solution.

Using the "Code sample" input (icon on the upper left just below "Reply to this discussion") will make the code easier to read. I made some changes in what you wrote to match what Mathematica requires so I might very well have made wrong changes. There needs to be spaces between some of the entries such as Fb23 needs to be "F b23" or "F * b23". I also made all of the capital letters into lowercase as sometimes uppercase might clash with an existing Mathematica function. And example of this is "E" which gets you 2.71828. And finally there is "(a112)/3" which I assume should either be "a11/3" or "a11 * 2/3". (I used "a11*2/3" in the example below.)

In[8]:= Solve[{c23 == c13 + a13, c22 == c12 + a12, c21 == c11 + a11, 
  b23 == 2 a a13, b22 == 2 a a12, b21 == 2 a a11, 
  b33 == b (2 a23 + b23), b32 == b (2 a22 + b22), 
  b31 == b (2 a21 + b21), c33 == a23 + b23 + c23, 
  c32 == a22 + b22 + c22, c31 == a21 + b21 + c21, a33 == -b33/2, 
  a32 == -b32/2, a31 == -b31 t/2, 
  a22/3 + b22/2 + c22 + f b23 == 
   e (2 a23 + b23), (2 a21)/3 + b21 + 2 c21 + f b22 == 
   e (2 a22 + b22), f b21 t == e (2 a21 t + b21 t), 
  a32/3 + b32/2 + c32 + h == g b33, (2 a31)/3 + b31 + 2 c31 == g b32, 
  0 == b31, a12/3 + c12 == d + 2 c a13, 2 c11 + a11 2/3 == 2 c a12, 
  0 == a11}, {a11, a12, a13, a21, a22, a23, a31, a32, a33, b21, b22, 
  b23, b31, b32, b33, c11, c12, c13, c21, c22, c23, c31, c32, c33}]


During evaluation of In[8]:= Solve::svars: Equations may not give solutions for all "solve" variables. >>

Out[8]= {{a11 -> 0, a12 -> 0, 
  a13 -> -((d e - b d g + e h)/(2 (c e - b c g - a b f g))), a21 -> 0,
   a22 -> 0, 
  a23 -> -((-a d e + a d f + a b d g + c h - a e h + a f h)/(
    2 (c e - b c g - a b f g))), a31 -> 0, a32 -> 0, 
  a33 -> (b (a d f + c h + a f h))/(2 (c e - b c g - a b f g)), 
  b21 -> 0, b22 -> 0, 
  b23 -> -((a d e - a b d g + a e h)/(c e - b c g - a b f g)), 
  b31 -> 0, b32 -> 0, 
  b33 -> (b (a d f + c h + a f h))/(-c e + b c g + a b f g), c11 -> 0,
   c12 -> -((a b d f g + c e h)/(c e - b c g - a b f g)), 
  c13 -> c23 - (-d e + b d g - e h)/(2 (c e - b c g - a b f g)), 
  c21 -> 0, c22 -> -((a b d f g + c e h)/(c e - b c g - a b f g)), 
  c31 -> 0, c32 -> -((a b d f g + c e h)/(c e - b c g - a b f g)), 
  c33 -> c23 - (a d e + a d f - a b d g + c h + a e h + a f h)/(
    2 (c e - b c g - a b f g))}}

So either I've incorrectly modified what you had or there is not a single solution.

POSTED BY: Jim Baldwin

Jie Li

Posted 9 years ago

Thank you a lot. So far, I got this, but how can I calculate c23? solve[{c23 == c13 + a13, c22 == c12 + a12, c21 == c11 + a11, c33 == a23 + b23 + c23, c32 == a22 + b22 + c22, c31 == a21 + b21 + c21, b23 == 2 a a13, b22 == 2 a a12, b21 == 2 a a11, b33 == b (2 a23 + b23), b32 == b (2 a22 + b22), b31 == b (2 a21 + b21), a33 == -b33/2, a32 == -b32/2, a31 == -b31 t/2, a11 == 0, 2 c a12 - a112/3 - 2 c11 == 0, d - a12/3 + c12 == 0, 2 e a21 + (e - f)b21 == 0, 2e a22 + (e - f)b22 - a212/3 - b21 - 2 c21 == 0, 2ea23 + (e - f)b23 - a22/3 - b22/2 - c22 == 0, gb21 == 0, gb32 - a312/3 - b31 - 2c31 == 0, g*b33 - h - a32/3 - b32/2 - c32 == 0}, {a11, a12, a13, a21, a22, a23, a31, a32, a33, b21, b22, b23, b31, b32, b33, c11, c12, c13, c21, c22, c23, c31, c32, c33}] {{a11 -> 0, a12 -> 0, a13 -> -((d e - b d g - e h)/(2 a b f g)), a21 -> 0, a22 -> 0, a23 -> -((-d e + d f + b d g + e h - f h)/(2 b f g)), a31 -> 0, a32 -> 0, a33 -> -((-d + h)/(2 g)), b21 -> 0, b22 -> 0, b23 -> -((d e - b d g - e h)/(b f g)), b31 -> 0, b32 -> 0, b33 -> -((d - h)/g), c11 -> 0, c12 -> -d, c13 -> c23 - (-d e + b d g + e h)/(2 a b f g), c21 -> 0, c22 -> -d, c31 -> 0, c32 -> -d, c33 -> c23 - (d e + d f - b d g - e h - f h)/(2 b f g)}}

POSTED BY: Jie Li

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