No wonder there is super-high barrier of entry. And that is why us mortals are just using numerical methods instead.
At the end of the day, most practitioners will work with just a bag of tricks, so if that is what the class teaches most of the time, I really don't mind at all. Yeah there is deeper theory to changes of variables and all that, but for an intro class that is aimed at a wide audience (physicists and any sort of applied math), then yeah some light theory and a bag of tricks is perfect. The students that need more will learn that theory later anyways in a second or even third class.
You don't have to sacrifice that much rigour either.
Not every class needs a narrative either. I agree that concrete examples to go with the abstract is more needed in mathematics, but differential equations isn't a victim of that.
I see other comments say that differential equations are victim to :
> why we are we learning this?
I don't think this really applies to differential equations at all to be honest. And people say the same complaints about learning scales in music or wtv. Just learn shit and figure out meaning later works too.
Reasonable people can probably disagree, and this might be biased by their personality and learning style.
I personally forgot every trick from my bags mere moments after walking out of my graduate exams. Later, when I got a job that required me to actually /do/ things, I (re-)learned the theory and tricks I needed for the task at hand. The courses that have had a real impact on me were not the "bag of tricks" ones, they were the ones that planted a small grain of understanding and/or appreciation in my brain. I think these are like seed crystals: They make later learning orders of magnitude more efficient and are far less tangible but far more important than "bags of tricks" learning.