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If you have some mathematical maturity (you can read notation, work to understand each equation, have the patience to pretty much understand each equation and e
by sampo 8y ago
If you have some mathematical maturity (you can read notation, work to understand each equation, have the patience to pretty much understand each equation and each page, because later ideas are built on understanding the previous ideas first), you can self-study a lot of mathematics. But almost everyone (mathematicians, physicists, engineers) who has that maturity, has probably taken a first course in differential equations during their undergrad studies.
So the number of people who have developed some mathematical maturity without ever taking a diff.eq. course is probably small. So we don't know much about these people and how the topic of differential equations appears to them.
But yes, it would be interesting to hear about experiences from someone with e.g. a strong background in theoretical computer science, probably including calculus but not including diff.eqs., who has self-studied a book on differential equations.
- mclehman 8y agoJust chiming in to say we do indeed exist and your example is pretty spot on, but I unfortunately have not self-studied diff.eqs. if I eventually do, I'll make sure to write something up about it. I'd like to think I have developed a hint of mathematical maturity (CS + late declared Math major), but I went more the algebra route and ended up never taking a diff.eq. course. The closest I came was in a complex analysis course with some motivating examples that assumed we'd have picked up some tricks in a diff.eq. course.
- kenjackson 8y agoI had self-studied this and then later studied under a teacher as well. I think the thing that I found hardest in my self-study was (and unfortunately this is about 25 years ago so my recollection might be a bit off) that it seemed like there was a lot written on just two equations (the heat equation and the wave equation). I didn't get why is 50 pages dedicated to one equation. Up until that point it felt like Calculus was about techniques to solve equations, and then it suddenly became mostly about how to solve these two equations (there was a third, but I can't recall what it was now), which never really resonated with me.
- sampo 8y agoHeat equation / diffusion equation (same thing): parabolic. Wave equation: hyperbolic. Laplace's equation: elliptic. Laplace's equation is the steady state (equilibrium) solution of heat equation when the boundary conditions (or anything) don't change in time.
- kenjackson 8y agoThanks. And yes, Laplace was the third one. I think even your simple categorization would have made things much simpler for me. Or maybe I was just an idiot as an undergrad. :-)