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MathGroup Archive 2006

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Re: solving an equation with sums

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67497] Re: solving an equation with sums
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Wed, 28 Jun 2006 03:52:25 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <e7qmi0$6lb$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

newbix at bk.ru wrote:
> Hello,
> 
> I try to solve the following equation:
> sum[2*(1-e^(-x*t[[i]])-F[[i]])*t[[i]]*e^(-x*t[[i]]),{i,1,10}]==0
> where t and F are lists. Each list has ten entries.
> I tried to solve this sum using the following command:
> ------------------------------------------------------
> Solve[2*(1 - \[ExponentialE]^(-x*0.1) - 2810)*0.1*\[ExponentialE]^(-x*0.1) + 
>       2*(1 - \[ExponentialE]^(-x*0.2) - 5411)*0.2*\[ExponentialE]^(-x*0.2) + 
>       2*(1 - \[ExponentialE]^(-x*0.3) - 8701)*0.3*\[ExponentialE]^(-x*0.3) + 
>       2*(1 - \[ExponentialE]^(-x*0.4) - 13130)*0.4*\[ExponentialE]^(-x*0.4) + 
>       2*(1 - \[ExponentialE]^(-x*0.5) - 17327)*0.5*\[ExponentialE]^(-x*0.5) + 
>       2*(1 - \[ExponentialE]^(-x*0.6) - 24899)*0.6*\[ExponentialE]^(-x*0.6) + 
>       2*(1 - \[ExponentialE]^(-x*0.7) - 31230)*0.7*\[ExponentialE]^(-x*0.7) + 
>       2*(1 - \[ExponentialE]^(-x*0.7) - 40006)*0.8*\[ExponentialE]^(-x*0.8) + 
>       2*(1 - \[ExponentialE]^(-x*0.8) - 59880)*0.9*\[ExponentialE]^(-x*0.9) + 
>       2*(1 - \[ExponentialE]^(-x) - 80017)*\[ExponentialE]^(-x) == 0, x]
> -----------------------------------------
> but I got the following error message:
> 
> Solve::"tdep": "The equations appear to involve the variables to be solved \
> for in an essentially non-algebraic way."
> 
> What's wrong? How can I solve this expression using Mathematica?
> 
> Thank you in advance!
> 
Use exact coefficients [1] and Reduce [2]. About Root objects, see [3].

In[1]:=
Reduce[Rationalize[(2*(1 - E^((-x)*0.1) - 2810)*0.1)/E^(x*0.1) +
      (2*(1 - E^((-x)*0.2) - 5411)*0.2)/E^(x*0.2) +
      (2*(1 - E^((-x)*0.3) - 8701)*0.3)/E^(x*0.3) +
      (2*(1 - E^((-x)*0.4) - 13130)*0.4)/E^(x*0.4) +
      (2*(1 - E^((-x)*0.5) - 17327)*0.5)/E^(x*0.5) +
      (2*(1 - E^((-x)*0.6) - 24899)*0.6)/E^(x*0.6) +
      (2*(1 - E^((-x)*0.7) - 31230)*0.7)/E^(x*0.7) +
      (2*(1 - E^((-x)*0.7) - 40006)*0.8)/E^(x*0.8) +
      (2*(1 - E^((-x)*0.8) - 59880)*0.9)/E^(x*0.9) +
      (2*(1 - E^(-x) - 80017))/E^x == 0], x]

Out[1]=
C[1] â?? Integers && (x ==

      10 (I Pi + 2 I Pi C[1] +

                            3       5       6       8
         Log[-Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                       10            11            12
              800165 #1   + 538911 #1   + 320044 #1   +

                       13            14           15
              218603 #1   + 149391 #1   + 86630 #1   +

                      16           17           18
              52518 #1   + 26100 #1   + 10821 #1   +

                     19
              2809 #1   & , 1]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 2]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 3]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 4]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 5]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 6]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 7]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 8]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 9]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 10]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 11]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 12]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 13]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 14]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 15]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 16]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 17]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 18]]) ||

     x == 10 (2 I Pi C[1] +

                           3       5       6       8
         Log[Root[10 + 9 #1  + 8 #1  + 7 #1  + 6 #1  +

                      10            11            12
             800165 #1   + 538911 #1   + 320044 #1   +

                      13            14           15
             218603 #1   + 149391 #1   + 86630 #1   +

                     16           17           18
             52518 #1   + 26100 #1   + 10821 #1   +

                    19
             2809 #1   & , 19]]))

Regards,
Jean-Marc

[1] http://documents.wolfram.com/mathematica/functions/Rationalize

[2] http://documents.wolfram.com/mathematica/functions/Reduce

[3] http://documents.wolfram.com/mathematica/functions/Root


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