Group Abstract

Message Boards

WOLFRAM COMMUNITY

6.6K Views

4 Replies

3 Total Likes

View groups...

Follow this post

Share this post:

GROUPS:

Mathematics Mathematica Algebra

Can this equation system be solved?

Fred Alstair

Posted 10 years ago

12500/13 * 13^(x/5)== y * 2 * 10^07 * 4^((y - 175)/25) * 2.76 * 10^07 * d * 0.03 y = Log[1.07, -((z/(6.510^9)) (1 - 1.07)/1.07)] z = 301 * d * 0.01 * 11400 * -(13^((x/5) + 1)/(1 - 13)) d > 0 x > 0 trying to find f(d)= some value of x if you replace d with a value it will give the right answer, but i cant find the formula for any d for example /.d -> 37878 give the right answer x -> 309.583 but trying for a general solution only give a solution where either d or x is negative

POSTED BY: Fred Alstair

4 Replies

Sort By:

Fred Alstair

Posted 10 years ago

thank you, that was really helpful, though before seeing your solution i did find one of my own though less elegant : pointList = Table[f = y210^074^((y - 175)/25)6.6510^04 d 0.03((d/33100)/25/810 + 1)((d/33100)/25/75 + 1); y = Log[1.07, -((z/(6.510^9))(1 - 1.07)/1.07)]; z = 58.82 ((d/33100)/25/810 + 1) d0.0111400-(13^((x/5) + 1)/(1 - 13)); V1 = Solve[{f == 12500/1313^(x/5)}, x]; {d, x /. V1[[2]]}, {d, 100, 50000, 100}] the code change a bit since i posted it there's also 2 point that appear to be wrong not sure how they got there

thank you, that was really helpful, though before seeing your solution i did find one of my own though less elegant :

pointList = 
 Table[f = 
   y*2*10^07*4^((y - 175)/25)*6.65*10^04 * 
    d *0.03*((d/33*100)/25/810 + 1)*((d/33*100)/25/75 + 1);
  y = Log[1.07, -((z/(6.5*10^9))*(1 - 1.07)/1.07)];
  z = 58.82* ((d/33*100)/25/810 + 1)*
    d*0.01*11400*-(13^((x/5) + 1)/(1 - 13));
  V1 = Solve[{f == 12500/13*13^(x/5)}, x];
  {d, x /. V1[[2]]}, {d, 100, 50000, 100}]

the code change a bit since i posted it there's also 2 point that appear to be wrong not sure how they got there
enter image description here

POSTED BY: Fred Alstair

Shenghui Yang

Shenghui Yang, WOLFRAM

Posted 10 years ago

The equation you are solving is transcendental so usually it does not have a simple solution from Solve function. Also once you have numeric coefficients in the equation, I think NSolve is used automatically. The reason that you have two solutions from your code is the Log function used for z works on complex plane or the negative arguments, you just encountered a complex root there. I modified the solver in your code with the FindRoot function so that the only real solution is guaranteed to be spotted on.

Use ReplaceAll (//.) to save some variables.

Clear["Global`*"]

f = y*2*10^07*4^((y - 175)/25)*2.76*10^07 * d *0.03 //.
   {y -> Log[1.07, -((z/(6.5*10^9))*(1 - 1.07)/1.07)], 
    z -> 301 *d*0.01*11400*-(13^((x/5) + 1)/(1 - 13))};

I use Log function to map the f and the equation that you are solving onto the logarithmic plane so that the solution converges fast.

h1[x_, d_] = 
 FullSimplify[f, 
  Assumptions -> {x \[Element] Reals, d \[Element] Reals}]
 (*80704. d^1.81958 E^(0.420437 x) Log[3.7414*10^-7 13^(x/5) d]*)

h2[x_, d_] = N@PowerExpand@(Log[h1[x, d]] - Log[12500/13*13^(x/5)])
 (*1.86506 - 2.56495 (-1. + 0.2 x) + 0.420437 x + 1.81958 Log[d] +  Log[-14.7986 + 0.51299 x + Log[d]]*)

(*with small x, the value of h2 can be complex that is why you have \two solutions*)
h2[10, 1000] (* result = 17.0893 + 3.14159 I*)

Given d=1000, we can find the root of h2 by Plot function and the intersection between the curve and the x axis: enter image description here