I just finished a Great Courses series about differential equations. This is kind of a big deal for me because in college, I changed my major away from Physics largely because the linear algebra course exhausted me and I looked at DiffEq coming up next and went "NOPE".
But the approach of this lecture series was highly visual, using a lot of diagrams and computer tools to show how the functions behaved. Turns out this fits my style of learning a lot more than trying to parse text equations. I was captivated by the course, and managed to follow the material all the way to the end. This is not to say that I could necessarily pass an exam, I generally didn't do the homework problems so I admit that my understanding of it isn't terribly rigorous, but the subject no longer intimidates me. I can even look at differential equations in physics (like Maxwell's equations for instance) and go "yeah I see what's going on there."
Moreover, there's a sort of... flavor... to math at this level that I find really interesting: One thing I realized at some point in the course is that a lot of differential equations are sort of "unsolvable", or at least they're far too complicated to be easily sorted out, compared to something like 2x - 1 = 7. And yet, there are still tools that can be used to understand a lot about them, like the large-scale shape of solutions, or the behavior at critical points, and that's enough to be able to deal with the equations usefully. It feels both cool to know, and also somehow profound.
is how much stuff doesn't have a solution
but also, how much you start to wonder what a "solution" is supposed to be
like, what is the square root of 2? that's not a number, not quite; it's a minimal recipe for how to get to a number, and it's written in a way that lets you know a lot of convenient properties of that number. if you want the actual number we know a bunch of ways to get as much precision out of it as you want, bit it does feel like something has been lost then
but integrals we can't evaluate and differential equations we can't solve also fit that definition, even if the list of convenient properties is much shorter
and then you find out that quintic and higher polynomials also have no general solutions in familiar operations, and while they can be found and written out, they use things called hypergeometric functions that are themselves just very general recipes for evaluating a number with as much precision as you want
and this is like 9th grade algebra and it turns out they quietly hid this from you the whole time, maybe to avoid instilling teenagers with the disillusionment of knowing that our teeny tiny mathematical language can only express the answers to a zero-measure slice of all possible questions
but if you look at it from the other direction, it's actually quite impressive that we managed to look out across the chaos of numbers and somehow eke out a couple tools that are useful for any of it at all
