Introduction to Differential Equations (2008) • Hacker News Board

Re: Introduction to Differential Equations (2008)

#21

Post by RivieraKid » Wed, Oct 10, 2018, 9:43 AM UTC

Can anyone compare this vs other resources to learn differential equations? I want to learn math roughly to the level of an undergrad engineering student, so I've looked at some Advanced Engineering Mathematics books (one by Zill, another by Kreyszig), both have mostly good reviews, and to be honest, Paul's Notes seem almost a level above in clarity and understandability. For example, compare the explanation of integrating factor and exact equations. The books typically explain exact equations via a total differential, which I'm kind of confused about... why not just say that the left part of the equation is a total derivative?

I can't decide whether to continue reading the Advanced Engineering Mathematics book or learn the topics it contains via Paul's Notes and other resources. My worry is that the reason Paul's Notes seem clearer is simply because they're more superficial.

What are some other good learning resources for advanced engineering undergrad math?

Re: Introduction to Differential Equations (2008)

#22

Post by posterboy » Wed, Oct 10, 2018, 9:43 AM UTC

freetime2 wrote: post #19

acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…

Using my own words, I'd say it was necessary though. One thing is the technique abstracted from the application, the other thing is a concept applying the technique. A fully abstract technique is awfully dry, so it's best taught connecting to other techniques. At that point, ODEs are a concept reduced to mathematics.

I found ODEs interesting only later when used in Linear Algebra, which could lead to differential algebra. But there are different approaches and not one is the true one, so what should an instructor do? You need calculus for ODEs, so much is true I guess.

Growth in hunter/food populations is interesting, too, plotting fox pop. over hare pop. Finally graphs can go in circles, not just in one direction along one axis.

Re: Introduction to Differential Equations (2008)

#23

Post by RivieraKid » Wed, Oct 10, 2018, 9:48 AM UTC

Sean1708 wrote: post #15

The course overview is here[0]. I haven't looked at how in-depth the later sections are, but I can't think of a topic related to differential equations that I used in my undergrad physics degree that isn't at least touched upon here. [0]: http://tutorial.math.lamar.edu/Classes/DE/DE.aspx

It doesn't include Green's functions (I don't really know what that is but some other undergrad diff eq books have them).

Re: Introduction to Differential Equations (2008)

#24

Post by dan00 » Wed, Oct 10, 2018, 9:50 AM UTC

freetime2 wrote: post #19

acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…

The "bag of tricks" seems to allow to efficiently test a lot of people by writing tests and scoring them.

Unfortunately it seems to be more about the score than about the learning.

Re: Introduction to Differential Equations (2008)

#25

Post by klrr » Wed, Oct 10, 2018, 10:00 AM UTC

sideproject wrote: post #16

Does anyone know websites or resources which explains how Diff Eq is used in Computer Science? I know it's used in a variety of areas in CS, but I really like to see or read well-explained tutorials or articles. e.g. what's finding area got to do with the topics in CS? what does 'area' correspond to?

I'm also interested in this, it comes up in a lot of EE courses (DSP, Control theory), but I've yet to see it used in a CS course.

Re: Introduction to Differential Equations (2008)

#26

Post by pbhjpbhj » Wed, Oct 10, 2018, 10:14 AM UTC

vonseel wrote: post #10

Wow, it's been a while. Does anyone actually use this stuff after college???

IMO that's a well made question, I think it's good to ask to.

https://news.ycombinator.com/item?id=18182657 is an example ITT WRT computer game writing.

Re: Introduction to Differential Equations (2008)

#27

Post by soVeryTired » Wed, Oct 10, 2018, 10:23 AM UTC

freetime2 wrote: post #19

acjohnson55 wrote: post #4

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it…

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…

I feel like Vladimir Arnold echoed a similar sentiment, but I can't find the quote. He was complaining that a first course in ODE and PDE tends to teach methods for finding exact solutions, even though equations with exact solutions are vanishingly rare in practice.

Or maybe it was Rota again, who knows?

Edit: nope, Rota again. Lesson one in your link :)

Re: Introduction to Differential Equations (2008)

#28

Post by jstewartmobile » Wed, Oct 10, 2018, 10:31 AM UTC

I think the schools spend far too much time on symbolic differentiation and integration. This limits the exercises to the kinds of toy problem that yield to those methods. Kids get sidetracked on solving anti-differentiation puzzles, while the fundamentals are relegated to those (largely useless) puzzles.

After 20 years of engineering--in almost every case--numerical methods have been the only way forward. In hindsight, a year-and-a-half long course to convey the fundamentals seems excessive.

Re: Introduction to Differential Equations (2008)

#29

Post by bigasscoffee » Wed, Oct 10, 2018, 10:40 AM UTC

used these sites all of the time during engineering. improper integrals, diff eq, etc.

these are the best notes ever if you missed class or a concept.

Re: Introduction to Differential Equations (2008)

#30

Post by enriquto » Wed, Oct 10, 2018, 10:55 AM UTC

soVeryTired wrote: post #27

freetime2 wrote: post #19

Earlier quoted context omitted.

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks…

I feel like Vladimir Arnold echoed a similar sentiment, but I can't find the quote. He was complaining that a first course in ODE and PDE tends to teach methods for finding exact solutions, even though equations with exact solutions are vanishingly rare in practice. Or maybe it was Rota again, who knows? Edit: nope, Rota again. Lesson one in your link :)

definitely in the spirit of Arnold

his book on ODE is otherworldly

the quote maybe comes from his pde book