Re: Why No Solution Using Solve?

• To: mathgroup at smc.vnet.net
• Subject: [mg33929] Re: [mg33902] Why No Solution Using Solve?
• From: Andrzej Kozlowski <andrzej at bekkoame.ne.jp>
• Date: Tue, 23 Apr 2002 07:14:11 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```The reason why solve gives the empty list of solutions is that these
equations only have solutions if the parameters satisfy certain
relationships, and there are no solutions valid for all values of the
parameters. Solve only looks for generic solutions, that is solutions
valid for all values of the parameters. The function to use in cases
like this is normally Reduce, but the relations involved here appear to
be so complicated that I doubt Reduce will arrive at an answer in a
reasonable time. To see what happens look at:

In[1]:=

Eliminate[{
m1 a1 == P1/ v1 - A1 Rho Cd (v1 + w)^2 /2 - m1 g Cr1 - m1 g S,
m2 a2 == P2/ v2 - A2 Rho Cd (v2 + w)^2 /2 - m2 g Cr1 - m2 g S,
m3 a3 == P3/ v3 - A3 Rho Cd (v3 + w)^2 /2 - m3 g Cr1 - m3 g S},
{ Cr1,S}
]

Out[1]=
2*m2*P1 == v1*(2*a1*m1*m2 - 2*a2*m1*m2 + A1*Cd*m2*Rho*v1^2 +
(2*m1*P2)/v2 - A2*Cd*m1*Rho*v2^2 + 2*A1*Cd*m2*Rho*v1*
w - 2*A2*Cd*m1*Rho*v2*w - A2*Cd*m1*Rho*w^2 +
A1*Cd*m2*Rho*w^2) && 2*m3*P1 ==
v1*(2*a1*m1*m3 - 2*a3*m1*m3 + A1*Cd*m3*Rho*v1^2 +
(2*m1*P3)/v3 - A3*Cd*m1*Rho*v3^2 + 2*A1*Cd*m3*Rho*v1*
w - 2*A3*Cd*m1*Rho*v3*w - A3*Cd*m1*Rho*w^2 +
A1*Cd*m3*Rho*w^2) && 2*m3*P2 ==
v2*(2*a2*m2*m3 - 2*a3*m2*m3 + A2*Cd*m3*Rho*v2^2 +
(2*m2*P3)/v3 - A3*Cd*m2*Rho*v3^2 + 2*A2*Cd*m3*Rho*v2*
w - 2*A3*Cd*m2*Rho*v3*w - A3*Cd*m2*Rho*w^2 +
A2*Cd*m3*Rho*w^2) && v1 != 0 && v2 != 0 && v3 != 0

So we have eliminated two of your solve variables, Cr1 and S and we get
a set of three quadratic equations in w (and three additional conditions
on v1,v2, and v3). So you can see that there is not going to be a
general solution. You can now take the three equations in w and
eliminate w from them to obtain a complicated set of conditions on the
parameters that has to be satisfied for solutions to exist. These
conditions are so long and awful I decided not to paste them in.
Getting the corresponding solutions is quite another matter!

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/

On Monday, April 22, 2002, at 01:57  PM, Tom Compton wrote:

> Why doesn't this give a solution?
>
> Solve[{
> m1 a1 == P1/ v1 - A1 Rho Cd (v1 + w)^2 /2 - m1 g Cr1 - m1 g S,
> m2 a2 == P2/ v2 - A2 Rho Cd (v2 + w)^2 /2 - m2 g Cr1 - m2 g S,
> m3 a3 == P3/ v3 - A3 Rho Cd (v3 + w)^2 /2 - m3 g Cr1 - m3 g S},
> {Cr1, S, w}
> ]
>
>
>
>
>
>

```

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