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

[xxxyyy]

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.

 

[scottdave]

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?

 

[Mark44]

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.

 

[xxxyyy]

Here are just simple images.

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