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Re: Why No Solution Using Solve?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg33930] Re: [mg33902] Why No Solution Using Solve?
*From*: Andrzej Kozlowski <andrzej at bekkoame.ne.jp>
*Date*: Tue, 23 Apr 2002 07:14:13 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
I need to make a correction to the last message. At the end of it I
suggested that while it is fairly easy to find the conditions the
parameters must satisfy for solutions to exist, it is much harder to
find the solutions themselves. In fact, it is quite "easy" to find
solutions to the equations, once we know that the parameters satisfy
the conditions given by elimination. The possible values of w will be,
of course, just the solutions of each of the the three quadratics in w
obtained by the first Eliminate and once these are known, Cr1 and S can
be found accordingly. It's probably easier for Mathematica to do this
"under human guidance" than directly by means of Reduce.
On Monday, April 22, 2002, at 11:04 PM, Andrzej Kozlowski wrote:
> 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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