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Re: 3 eqns 3 unknws

  • To: mathgroup at smc.vnet.net
  • Subject: [mg40950] Re: [mg40930] 3 eqns 3 unknws
  • From: Bobby Treat <drmajorbob+MathGroup3528 at mailblocks.com>
  • Date: Fri, 25 Apr 2003 08:04:55 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Mathematica will not solve for {x,y,z} unless it can do so for ALL 
values of the other parameters without conditions on them.  The form 
and even NUMBER of different solutions depends on a, b, and c in a 
complex way, so there's no reasonable way to state a general solution 
even if we could find it.

If you are willing to specify a, b, and c before solving, you have a 
reasonable chance.

You don't want to be using D as a variable name.  It is the 
differentiation operator.  In general, it's not recommended to start 
variables with capital letters.  Also, A and P in your problem always 
appear multiplied together, so I would replace them with a single 
variable (ap, for instance) equal to their product.

eqns = {a*P*A*
  x^(a - 1)*y^b*z^c == r, b*P*A*x^a*y^(b - 1)*z^c == w, c*P*A*x^
      a*y^b*z^(c - 1) == D/(z^2)} /. {P -> ap/A, D -> d, a -> 1,
      b -> 2, c -> 3}

Solve[eqns, {x, y, z}]

{{x -> (4*ap*d^3)/(27*r^2*w^2),
   y -> (8*ap*d^3)/(27*r*w^3),
   z -> (9*r*w^2)/(4*ap*d^2)}}

This gets MUCH more complicated if a, b, and c are not integers, and 
that's why Mathematica won't even attempt the general problem.  Try 
this one, for instance:

eqns = {a*P*A*
  x^(a - 1)*y^b*z^c == r, b*P*A*x^a*y^(b - 1)*z^c == w, c*P*A*x^
      a*y^b*z^(c - 1) == D/(z^2)} /. {P -> ap/A, D -> d, a -> 0.1,
       b -> 2, c -> 3}
Solve[eqns, {x, y, z}]

(19 solutions, most of them Complex)

Change a to 0.2 and there are only 9 solutions.  Change it to 0.3, and 
there are 17 solutions.  Increase b or c, and it gets worse.

Bobby

-----Original Message-----
From: Richard Cochinos <richard at theory.org>
To: mathgroup at smc.vnet.net
Subject: [mg40950] [mg40930] 3 eqns 3 unknws


I cant seem to get mathematica to solve a set of algebraic equations, 
any
suggestion? I'm looking for x,y,z ; p,r,w,d are all constants.

"In[1]:=Solve[{a*P*A*x^(a - 1)*y^b*z^c == r, b*P*A*x^a*y^(b - 1)*z^c == 
w,
    c*P*A*x^a*y^b*z^(c - 1) == D/(z^2)}, {x, y, z}]

Solve::"tdep": "The equations appear to involve the variables to be 
solved
for in an essentially non-algebraic way."

Out[1]:= \!\(Solve[{a\ A\ P\ x\^\(\(-1\) + a\)\ y\^b\ z\^c == r,
      A\ b\ P\ x\^a\ y\^\(\(-1\) + b\)\ z\^c == w,
      A\ c\ P\ x\^a\ y\^b\ z\^\(\(-1\) + c\) == D\/z\^2}, {x, y, z}]\)"

Well, the last part doesn\t translate to text proper, but it just
reitterates the equations.

  r.j.c.//richard at theory.org


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