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

In summary, The conversation discusses a problem with a system and the difficulty the person is having in solving it. They request help with obtaining the equations and solution from a pdf file, but the conversation becomes more complex as the person tries to explain their attempts and the author's explanation. Ultimately, the person provides simple images of the equations and solution, expressing uncertainty about their accuracy.
  • #1
xxxyyy
14
2
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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  • #2
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?
 
  • #3
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.
 
  • #4
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
 

1. What is a limit in a 2 equation system of 1st order ODEs?

A limit in a 2 equation system of 1st order ODEs refers to the behavior of the system as the independent variable approaches a certain value. It is the value that the dependent variables tend towards as the independent variable gets closer and closer to the specified value.

2. Why is taking a limit important in solving 1st order ODEs?

Taking a limit is important in solving 1st order ODEs because it allows us to determine the behavior of the system at a specific point, even if the system is not defined at that point. It also helps us to understand the overall behavior of the system and make predictions about its future behavior.

3. How do you take a limit in a 2 equation system of 1st order ODEs?

To take a limit in a 2 equation system of 1st order ODEs, we first need to determine the independent variable that we want to approach. Then, we substitute this value into the equations and simplify as much as possible. Finally, we take the limit of the resulting expressions to find the behavior of the system at that specific point.

4. What are some common techniques for taking limits in 1st order ODEs?

Some common techniques for taking limits in 1st order ODEs include using L'Hôpital's rule, factoring and simplifying, and using known limits and properties of limits. It is also important to understand the behavior of basic functions such as polynomials, exponential, and trigonometric functions in order to accurately take limits in ODEs.

5. Can taking a limit help us solve a 2 equation system of 1st order ODEs?

Yes, taking a limit can help us solve a 2 equation system of 1st order ODEs by providing information about the behavior of the system at a specific point. This can help us determine initial conditions or make predictions about the behavior of the system at other points. Additionally, taking a limit can also help us simplify the equations and make them easier to solve.

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