System of Implicit Non-Linear First Order ODEs

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Discussion Overview

The discussion revolves around a complex system of implicit non-linear first-order ordinary differential equations (ODEs). Participants are seeking general solutions and exploring various approaches to simplify the equations, including the use of intrinsic coordinates.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses the need for ideas on a general solution for a messy system of differential equations involving a function p(t) and a constant A.
  • Another participant suggests rewriting a term involving the arctangent function to simplify the equations, proposing a transformation to facilitate further analysis.
  • A different participant proposes switching to intrinsic coordinates (s, ψ) to reformulate the system, leading to conditions on the derivatives and the constant A.
  • There is a mention that if A = 0, then certain simplifications can be made, allowing ψ to be an arbitrary function of t.
  • Some participants note issues with the rendering of LaTeX code, which affects the clarity of the mathematical expressions being discussed.
  • One participant suggests that the first solution might involve setting specific derivatives to zero, leading to a straightforward solution.

Areas of Agreement / Disagreement

Participants are exploring different methods to approach the problem, and while some suggestions are made, there is no consensus on a single solution or method. Multiple competing views and approaches remain present in the discussion.

Contextual Notes

Participants express uncertainty regarding the clarity of mathematical expressions due to LaTeX rendering issues, which may affect the understanding of the proposed solutions.

CSteiner
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I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution?

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sin%20%5Cleft%20%5B%5Carctan%20%5Cfrac%7B%5Cdot%7By%7D%7D%7B%5Cdot%7Bx%7D%7D%20%5Cright%20%5D-At.gif


p(t) is a function of t, and A is a constant.
 
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CSteiner said:
I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution?

%5Ccos%20%5Cleft%20%5B%5Carctan%20%5Cfrac%7B%5Cdot%7By%7D%7D%7B%5Cdot%7Bx%7D%7D%20%5Cright%20%5D.gif

sin%20%5Cleft%20%5B%5Carctan%20%5Cfrac%7B%5Cdot%7By%7D%7D%7B%5Cdot%7Bx%7D%7D%20%5Cright%20%5D-At.gif


p(t) is a function of t, and A is a constant.
An idea would be to rewrite ##\cos \arctan \frac{\dot{y}}{\dot{x}}## as ##\frac{\dot x}{\sqrt{\dot x^2+\dot y^2}}##, and similar for the other equation. Then try to simplify it a bit.
 
Didn't think of that, thanks! I'll keep working on it.
 
CSteiner said:
I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution?

%5Ccos%20%5Cleft%20%5B%5Carctan%20%5Cfrac%7B%5Cdot%7By%7D%7D%7B%5Cdot%7Bx%7D%7D%20%5Cright%20%5D.gif

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p(t) is a function of t, and A is a constant.

You may be best served by switching to intrinsic coordinates (s, \psi) where <br /> \dot x = \dot s \cos\psi, \\<br /> \dot y = \dot s \sin \psi. Your system is then <br /> (\dot s - p) \cos \psi = 0 \\<br /> (\dot s - p) \sin \psi = -At. Now either \dot s = p or \cos \psi = 0. The first is impossible unless A = 0. The second requires that \dot x = 0 and \dot y = \pm p - At.

If A = 0 then \dot s = p and \psi can be an arbitrary function of t.
 
pasmith said:
You may be best served by switching to intrinsic coordinates (s, \psi) where <br /> \dot x = \dot s \cos\psi, \\<br /> \dot y = \dot s \sin \psi. Your system is then <br /> (\dot s - p) \cos \psi = 0 \\<br /> (\dot s - p) \sin \psi = -At. Now either \dot s = p or \cos \psi = 0. The first is impossible unless A = 0. The second requires that \dot x = 0 and \dot y = \pm p - At.

If A = 0 then \dot s = p and \psi can be an arbitrary function of t.

Sorry, but it looks like something went wacky with your latex code. Could you retype it? It's hard to understand what your saying.
 
CSteiner said:
Sorry, but it looks like something went wacky with your latex code. Could you retype it? It's hard to understand what your saying.
It takes time before equations render correctly but eventually they do. What are you seeing? Try a different browser or device.
Basically he says to substitute ##\dot x = \dot s \cos\psi##, ##\dot y = \dot s \sin \psi##. Then it's very simple to arrive at a solution.

BTW, perhaps the first solution should have been ##\dot x=0##, ##\dot y=p\pm At##.
 

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