I used the perturbation method on some DE's, and compared terms of to get the following equations:

I've then been told that I can examine this system for a new rescaling in time:

Letting

So using the chain rule

Which I can substitute into the above equations to get:

So my question to any knowledgeable person is what is the point of re-scaling onto a new time/what does it show/can I now use this to show something? Quite confused!

Thanks in advance!