Help with some fiddly algebra around a 1st order ODE

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    Algebra Ode
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Discussion Overview

The discussion revolves around solving a first-order ordinary differential equation (ODE) related to a boat coasting with resistance proportional to velocity raised to a power, \( V^n \). Participants are attempting to derive expressions for velocity \( V(t) \) and position \( x(t) \) while encountering discrepancies between their results and those presented in a textbook.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the ODE \( m\frac{dV}{dt} = -kV^n \) and claims to have solved it, arriving at an expression for \( V(t) \) that matches the textbook for the general case of \( n \).
  • The same participant expresses confusion over a simplification of the exponent in their derived expression for \( V(t) \), questioning whether it affects the outcome.
  • Another participant points out a potential error in the simplification, stating that the equality \( \frac{1}{1-n} = n-1 \) is not valid, particularly noting that \( n=1 \) is problematic for the left-hand side.
  • A third participant reflects on their own confusion regarding algebraic manipulations, specifically mixing up the notation for powers, and questions the possibility of simplifying the exponent further.
  • A fourth participant reassures that algebra mistakes are common and agrees that the exponent cannot be simplified, suggesting that the original expression must be retained.

Areas of Agreement / Disagreement

Participants express differing views on the validity of algebraic simplifications, particularly regarding the exponent in the expressions derived from the ODE. There is no consensus on the correct simplification or the implications of the discrepancies with the textbook answers.

Contextual Notes

Participants have noted potential misunderstandings in algebraic manipulation and the implications of specific values of \( n \) on the validity of their expressions. The discussion highlights the complexity of handling exponents in the context of differential equations.

ognik
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Frustratingly although I can solve the ODE, I am getting a different answer to the book. Now going in circles so would appreciate a fresh pair of eyes.

The ODE (for a boat coasting with resistance proportional to $V^n$) starts as $ m\frac{dV}{dt} =-kV^n $ Find V(t) and x(t), V(0) = $V_0$

I solved the general case for n as $ V={V}_{0}[1+(n-1)\frac{k}{m}t{V}_{0}^{n-1}]^{\frac{1}{1-n}} $, which happily agrees with the book. But I simplified the power of the [...] term to $ V={V}_{0}[1+(n-1)\frac{k}{m}t{V}_{0}^{n-1}]^{n-1} $ - can't see that would make a difference?

Then, integrating again to get x(t), I let the term [...] = u, then $ dt=\frac{m}{(n-1)k{V}_{0}^{n-1}}du $

and $ x(t)=\frac{m{V}_{0}}{(n-1)k{V}_{0}^{n-1}}\int{u}^{n-1}du = \frac{m{V}_{0}}{(n-1)k{V}_{0}^{n-1}}\frac{{u}^{n}}{n} = \frac{m{V}_{0}^{2-n}}{kn(n-1)}[...]^n $

But the book shows $ \frac{m{V}_{0}^{2-n}}{k(2-n)}(1 - [...]^\frac{n-2}{n-1}) $?
 
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ognik said:
Frustratingly although I can solve the ODE, I am getting a different answer to the book. Now going in circles so would appreciate a fresh pair of eyes.

The ODE (for a boat coasting with resistance proportional to $V^n$) starts as $ m\frac{dV}{dt} =-kV^n $ Find V(t) and x(t), V(0) = $V_0$

I solved the general case for n as $ V={V}_{0}[1+(n-1)\frac{k}{m}t{V}_{0}^{n-1}]^{\frac{1}{1-n}} $, which happily agrees with the book. But I simplified the power of the [...] term to $ V={V}_{0}[1+(n-1)\frac{k}{m}t{V}_{0}^{n-1}]^{n-1} $ - can't see that would make a difference?

Why is
$$\frac{1}{1-n}=n-1?$$
That's the only thing you've changed in the expression. But this equality is manifestly not true. If for no other reason, $n=1$ is not in the domain of the LHS, but it is in the RHS.
 
How I can get the hard stuff done and then do something like that is beyond my understanding... maybe it's time to take a break, I think I mixed up $a^{-n} $ as $ a^{\frac{1}{n}}$ instead of $\frac{1}{{a}^{n}} $ - there is no simplification of a power like $ \frac{1}{1-n}$ is there?
 
Hey, it happens to the best of us. Algebra mistakes are by far the most common I run across, even in advanced math courses.

No, I don't think you can simplify the exponent at all. I think you're stuck with it.

Cheers!
 

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