/sci/ - Science & Math

Doing a course on some advanced topics in partial differential equations without having taken an "intro" to partial differential equations course due to some scheduling nonsense that means if I want to graduate on time I need to do well in this course. Done linearization and systems of ODEs before as well some "Turing morphogenesis" problems, so I might be okay but I have a particular question that may be very dumb:

What exactly is a rescaling? We do it a lot and I understand on some level that it's just convenient algebraic manipulation but it's got something to do with the method of dominant balance, getting your nonlinear term to scale in the same way as your linear term, but I don't quite understand exactly why it works out or why it's even justified to say "just let this be this times [math]\varepsilon ^2[/math] so that our linear term will be [math]\varepsilon[/math] so you can see the algebra works out"

Anyone know what I'm talking about or should I bust out some specific examples? I'm just looking for some kind of reading material or way of thinking about it beyond "just shut up and do what the professor says." Might not even be an issue insofar as on the homework he seems to have given us hints as to what rescalings to choose, but I wish I understood on some kind of higher level.