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