How to solve V=B+CT(V^N) for V?

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Discussion Overview

The discussion revolves around solving the equation V=B+CT(V^N) for V. Participants explore the nature of the equation, potential solutions, and the challenges associated with finding a closed-form solution. The scope includes mathematical reasoning and problem-solving approaches.

Discussion Character

  • Mathematical reasoning
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant seeks assistance in isolating V from the equation V=B+CT(V^N) and presents a proposed solution.
  • Another participant questions the validity of the proposed solution, suggesting that there may not be a closed-form solution for the equation.
  • A different participant points out that the equation can be transformed into a polynomial form, indicating that there is no general formula for solving polynomial equations of degree greater than 5.
  • Concerns are raised about the proposed solution not being valid in specific cases, providing a counterexample to illustrate the point.
  • Another participant suggests that if the equation relates to thermodynamics, an asymptotic solution may be sought for large values of V and N, acknowledging the difficulty of finding an exact solution.

Areas of Agreement / Disagreement

Participants express disagreement regarding the existence of a closed-form solution for the equation. Some believe that the proposed solution is incorrect, while others suggest that approximate solutions may be possible under certain conditions.

Contextual Notes

The discussion highlights the limitations of solving polynomial equations and the specific conditions under which approximate solutions might be applicable. There is an acknowledgment of the complexities involved in the equation without resolving them.

thiago.omena
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Does anyone know how to solve V=B+CT(V^N) for V?

The answer is V=[B^(1-N)+C(N-1)T]^(1/(1-N))
I don't understand how to get there.
Please help me out. Thank you
 
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Hey thiago.omena and welcome to the forums.

Are these matrices or just plain old numbers?
 
chiro said:
Hey thiago.omena and welcome to the forums.

Are these matrices or just plain old numbers?

In the question? They are just variables. I am trying to isolate the V.
 
thiago.omena said:
Does anyone know how to solve V=B+CT(V^N) for V?
The answer is V=[B^(1-N)+C(N-1)T]^(1/(1-N))
I don't understand how to get there.
What makes you think that's the answer? It doesn't look right to me. In fact, I do not believe there is a solution in closed form.

Btw, it's better to use the formatting this site provides. If you click Go Advanced you'll see a toolbar with e.g. X2 for superscript. Select the text to be superscripted and click the icon:
V=B+CT(VN)
 
V= B+ CTV^N is the same as the polynomial equation CTV^N- V+ B= 0. There is NO general formula for solving a polynomial equation of degree greater than 5. In fact, it was proved about a hundred years ago that there cannot be a formula (involving only roots of numbers) for such equations.

And it is easy to show that your V=[B^{1-N}+C(N-1)T]^(1/(1-N)) is NOT a general solution to the equation. In the case that C= T= 1, B= -2 and N= 2, the equation becomes V= -2+ V^2 which is the same as V^2- V- 2= (V- 2)(V+ 1)= 0 which has roots 2 and -1. But your formula gives V= [(-2)^{-1}+ (1)(-1)(1)]^{1/-1}= [-1/2+ -1]^{-1}= [-3/2]^{-1}= -2/3 which is NOT a solution.
 
thiago.omena said:
Does anyone know how to solve V=B+CT(V^N) for V?

The answer is V=[B^(1-N)+C(N-1)T]^(1/(1-N))
I don't understand how to get there.
Please help me out. Thank you

Is this a thermodynamic equation? If it is, you're likely looking for an asymptotic solution for V >> 1 and N >> 1. As has been said, solving the equation exactly is not possible in general, but an approximate solution in the large V and/or N limits may be possible, and is perhaps the answer you claim to have.
 

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