Take a limit in this 2 equation system of 1st order ODEs

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Discussion Overview

The discussion revolves around a system of first-order ordinary differential equations (ODEs) presented in a document. Participants are attempting to understand how to take a limit in the context of these equations, specifically for the case where κ1 is much smaller than κ2. The focus is on the mathematical manipulation of the equations rather than their physical implications.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in extracting the equations from a PDF and seeks assistance.
  • Another participant requests a different format for the equations and suggests posting an attempt at a solution.
  • A third participant notes that while they could access the file, the author's explanation of eliminating a variable from the equations lacks clarity, suggesting that decoupling the system might be necessary for finding a solution.
  • A later reply provides images of the equations and presents a potential solution for one of the variables, while cautioning against focusing on other aspects of the document.

Areas of Agreement / Disagreement

Participants have not reached a consensus on how to approach the problem, and there are multiple competing views on how to interpret and solve the system of equations.

Contextual Notes

There are limitations in the clarity of the author's explanation regarding the elimination of variables, and the participants have not fully resolved the mathematical steps necessary for the solution.

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Hello,
I'm having a problem with this system. Ignore the physics.
I have the feeling it should be tremendously easy... but I can't figure it out.
I don't know how to extract it from the pdf so I'll post just the these 2 pages.

https://ufile.io/39ovq

The equations are (1.14) and (1.15), the definition of Δ is in (1.13) (don't bother with the integral) for α and β go from 1 to 2.
The limit is for κ1 << κ2. n1 and n2 are just integer numbers.
The solution is in (1.16).
Thanks for the help.
 
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I cannot figure out how to display your pages. Can you try a different site like imgur, with just jpeg of the problem you are having trouble with. Can you also post your attempt at the solution, as well?
 
I was able to open the file, but the author's description of what he did isn't very enlightening. He mentions eliminating ##c_2## from equations 1.14 and 1.15, but doesn't give any insight into how he did this. The only thing I can think of is decoupling the system of differential equations to get the solution of the system (which is pretty involved), and then making additional assumptions to get the approximations he shows.
 
Here are just simple images.

https://ibb.co/vsqnJW3
https://ibb.co/yNMNgK3

vsqnJW3

yNMNgK3

and here the exact solution (if I didn't do any mistakes...) of the system for c1, (1.18), don't bother with the other junk.

https://ibb.co/pRDxKhc

pRDxKhc
 

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