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