16 ms·

A First Course in Differential Equations for Scientists and Engineers

514 points by lainon 8y ago

qwerty_danny 8y ago
IMO, Paul's notes http://tutorial.math.lamar.edu/Classes/DE/DE.aspx http://tutorial.math.lamar.edu/Classes/DE/DE.aspx are a lot easier to use.
ericand 8y ago
Collection of PDFs, more/less like a textbook. Looks worthwhile.
gattr 8y ago
On a related note, I can recommend Stanley J. Farlow's "Partial Differential Equations for Scientists and Engineers", which I bought around 2004 to better understand a fluid simulation paper I was trying to implement. (My Maths course ended with linear algebra and ODEs). Very good read with nice physics examples.
Luc 8y ago
Lately I have been looking for a practical (not too much theory) book on how to model problems with differential equations, with lots of examples. I imagine I'd plug them into Wolfram Alpha to get a solution. Does anyone have recommendations?
- sampo 8y ago
  Almost all textbooks named "Mathematical Modeling" should have chapters on modelling continuous change with differential equations, hopefully also a chapter on how to model systems of interacting variables with a system of differential equations. Then also textbooks with names like "Mathematical Models in Biology" or "Mathematical Biology" should have chapters on population growth, a two-species predator-prey model, diseases and epidemiology, and sometimes also a chapter on chemical kinetics (these are all modelled with differential equations).
  - Luc 8y ago
    Thank you. Knowing the right search terms will help a lot.
- schuetze 8y ago
  You can also look into "system dynamics" software, which allows for the modeling of complex systems with very little user-facing mathematics. Dana Meadows wrote a book called "Thinking in Systems," which is a great intro to this subject.¹ I personally have used Stella by isee systems, which will output an equation from your model if you are so inclined, but I think there is cheaper/free software out there that will do similar types of modeling. 1. https://wtf.tw/ref/meadows.pdf https://wtf.tw/ref/meadows.pdf
  - Luc 8y ago
    Thanks, that looks very interesting!
spapas82 8y ago
We used the Boyce and DiPrima book back when I was an electrical engineering student (Elementary Differential Equations and Boundary Value Problems https://www.amazon.com/Elementary-Differential-Equations-Boundary-Problems/dp/0471089559 https://www.amazon.com/Elementary-Differential-Equations-Bou...). I remember it was an excellent book with many great examples and correlation with physics topics like mechanics, waves etc. Too bad our other mathematics books weren't at this high standard :/ Also, I really don't think that such topics can be self-studied. It seems really difficult to me to understand such topics without a teacher!
- jamescostian 8y ago
  > I really don't think that such topics can be self-studied Can you explain this? What makes something extremely difficult to be studied without a teacher? (I took calc in high school and never took diff eq, so my knowledge in this specific domain is basically zero)
  - 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 ago
      Just 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.
gmiller123456 8y ago
While the linked course seems a lot more thorough, I took the Udacity "Differential Equations in Action" [1] course, which I found very well done. For the homework you write Python programs to compute things like gravitational slingshots, modeling epidemics, wildfires, and the n-body problem. [1] https://www.udacity.com/course/differential-equations-in-action--cs222 https://www.udacity.com/course/differential-equations-in-act...
- sampo 8y ago
  While the linked course notes (I looked at the pdf, definitely good textbook quality and not just course notes, really) are absolutely more thorough in the classical undergrad diff.eq. topics, the Udacity course touches some further topics (stiffness, implicit methods, control theory) that the course notes don't.
- nraynaud 8y ago
  Can we get to the text and videos without all the school-ish crud? (starting a course, timelines etc.)
  - gmiller123456 8y ago
    Yes [1]. I haven't taken a Udacity course in quite a while, but at the time all of their videos were hosted on Youtube, so you should be able to find a lot of their courses just on Youtube. For the homework, I remember there was an include file that they didn't provide the source code for, but a Google search turned up a version someone wrote for using it locally. Usually it's pretty obvious when you've solved the problem and have the correct code, but to make absolutely sure, you have to upload your code to their server for verification. But you can take the whole class without ever visiting Udacity. [1] https://www.youtube.com/watch?v=8cLXVG2Q6D4&list=PLAwxTw4SYaPlT4WYwf_SzNA9FV0qpGE5A https://www.youtube.com/watch?v=8cLXVG2Q6D4&list=PLAwxTw4SYa...
    - nraynaud 8y ago
      thanks!
alexibm 8y ago
Look at Arnold's book on Diff, it is harder then others, but has absolutely different outlook.
- selimthegrim 8y ago
  That book is totally mindblowing, and obliterates artificial boundaries between physics and mathematics. It also (in the older Dover editions) had a cover where the phase portrait on the front looked like two angry eyes glaring at you that you hadn't learned enough math yet.
  - Koshkin 8y ago
    Well, modern theoretical physics seems to be nothing but mathematics (mostly advanced differential geometry and group theory). And this is a good thing, as it is the sign of how far along the subject has gone in its evolution. The theory of differential equations also reduces to differential geometry, which is what Arnold’s book seems to be striving to show, as does his other excellent text on mathematical principles of classical mechanics.
- ced 8y ago
  I can only find the first edition on Amazon, but there's apparently a third edition? http://www.matem.unam.mx/ernesto/EDO/Ar-EDO.pdf http://www.matem.unam.mx/ernesto/EDO/Ar-EDO.pdf https://www.amazon.com/Ordinary-Differential-Equations-MIT-Press/dp/0262510189/ref=sr_1_sc_1?s=books&ie=UTF8&qid=1536220865&sr=1-1-spell&keywords=Ordinary+Differential+Equations+arnol https://www.amazon.com/Ordinary-Differential-Equations-MIT-P...
  - gjm11 8y ago
    I think this is the third edition: https://www.amazon.com/x/dp/3540345639 https://www.amazon.com/x/dp/3540345639
bgorman 8y ago
When I was in college I supplemented my learning with Notes on Diffyqs, which I thought explained things much clearer and concisely than my assigned textbook. It is available for free and has tools for modeling systems in Octave/Matlab. https://www.jirka.org/diffyqs/ https://www.jirka.org/diffyqs/
TomMckenny 8y ago
On a related note, how do you take notes for math subjects with their multiline integrals and sumation and subscripts, division etc? I'm very much attached to recording every thing in simple text editors. Is there a "Notepad" or "TextEdit" for mathematical notation?
- rhaps0dy 8y ago
  you can use LaTeX notation and then render it with something. For example MathJax, or emacs's org-preview-latex-toggle to show the rendered equations in the same buffer you write in.
- schitzapplebits 8y ago
  Don’t. Pay attention, write down the things that aren’t in the text book. And then learn to read the text book. It’s a hugely valuable skill.
  - thanatropism 8y ago
    I'm easily distracted, particularly from internal digressions. Taking notes helps me hold on to the thread.
- takk309 8y ago
  I could not imagine trying to take notes with a computer in a math heavy subject. I had to use paper for everything during my undergrad in Civil Engineering. However, if you already have a computer with you, record the lecture and review it later.
  - mclehman 8y ago
    Just as a partial counterpoint to this and overall agreement with everyone else who responded, I've had success taking realtime notes on my laptop in a math class. It took a bit of practice and I'm sure I can't do it anymore, but it's not insurmountable by any means. I used latex and made liberal use of keyboard and software macros to do it, and one of the tricks was to realize that if I needed a quick-to-type way to typeset new thing X, I should just pretend I had such an implementation and make up its command on the spot. At my leisure, I could write up a conforming latex command that worked with all the notes I'd taken in realtime. That said, I've since come to realize that math notes don't help me as much as they seem to help others. I have greater success primarily listening during class and leaning on the textbook as well as online resources outside of class. I do second the use of emacs to handle the latex, but I don't think that realtime rendering is particularly important in a notes setting.
miles_matthias 8y ago
This was my favorite course in college. Super beneficial materials. On a side note, one of the reasons I loved this course was because it was online and only had 2 tests with no other assignments. The professor allowed you to schedule office hours any time you needed, but the course setup was sweet for self-studiers like me. Here's the book, Chapters 1-10 are on the midterm, Chapters 10-20 are on the final. No homework busy work, no other tests. Just 2 exams. Go.
mindcrime 8y ago
Huh... didn't expect to find a UNC-Wilmington link on HN today. I vaguely remember Dr. Herman. I think we used one of his other books in a class, or maybe he taught one of my classes. This brings back memories.
sizzzzlerz 8y ago
I took DiffEq as an EE undergrad and, to my surprise, aced the course. For some reason, the concepts just resonated with me unlike other math courses where I had to work to gain an understanding of the material. A few years later, after graduating, I was taking courses leading to an MSEE and Partial DiffEq was offered as an option that I took. Thinking it was going to be similar to regular DE, I was soon in way over my head. Totally different material and concepts. I did end up passing the course but I can honestly state that I don't think I deserved it.
tzs 8y ago
Speaking of differential equations, an oldie (1997) but goodie, "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota: https://web.williams.edu/Mathematics/lg5/Rota.pdf https://web.williams.edu/Mathematics/lg5/Rota.pdf
- gp7 8y ago
  > 4. Teach Changes of Variables As someone who has basically no formal training in mathematics outside what's required for undergraduate computer science this is a big one. The first time I saw it I was blown away. Everyone who knows of it seems to treat it as natural as breathing, and not worth the exposition
deleted 8y ago
[deleted]
fizixer 8y ago
The usage of terms 'initial value' and 'boundary value' is a massive failure of mathematics education. 'Initial' refers to time-like variables, and 'boundary' refers to space-like variables. There is no notion of time in mathematics. There is only a notion of space, due to geometry. I had the longest time coming to grips with the question, "mathematically what is the difference between initial value and boundary value?" only to realize the distinction is meaningless in mathematics. It's a relic of the past when differential equations were studied under physics, where time and space are a huge part of the conceptual foundation. Sometimes I wonder how much progress we would make in education if we didn't confuse the heck of our students in the name of convention and historical baggage.
- ASipos 8y ago
  Since time is 1D while space may have more dimensions, initial values are connected to ODEs while boundary ones with PDEs. Their behavior is quite different.

514 points by lainon 8y ago

IMO, Paul's notes http://tutorial.math.lamar.edu/Classes/DE/DE.aspx http://tutorial.math.lamar.edu/Classes/DE/DE.aspx are a lot easier to use.

ericand 8y ago

Collection of PDFs, more/less like a textbook. Looks worthwhile.

gattr 8y ago

On a related note, I can recommend Stanley J. Farlow's "Partial Differential Equations for Scientists and Engineers", which I bought around 2004 to better understand a fluid simulation paper I was trying to implement. (My Maths course ended with linear algebra and ODEs). Very good read with nice physics examples.

Luc 8y ago

Lately I have been looking for a practical (not too much theory) book on how to model problems with differential equations, with lots of examples. I imagine I'd plug them into Wolfram Alpha to get a solution. Does anyone have recommendations?

sampo 8y ago
Almost all textbooks named "Mathematical Modeling" should have chapters on modelling continuous change with differential equations, hopefully also a chapter on how to model systems of interacting variables with a system of differential equations. Then also textbooks with names like "Mathematical Models in Biology" or "Mathematical Biology" should have chapters on population growth, a two-species predator-prey model, diseases and epidemiology, and sometimes also a chapter on chemical kinetics (these are all modelled with differential equations).
- Luc 8y ago
  Thank you. Knowing the right search terms will help a lot.
schuetze 8y ago
You can also look into "system dynamics" software, which allows for the modeling of complex systems with very little user-facing mathematics. Dana Meadows wrote a book called "Thinking in Systems," which is a great intro to this subject.¹ I personally have used Stella by isee systems, which will output an equation from your model if you are so inclined, but I think there is cheaper/free software out there that will do similar types of modeling. 1. https://wtf.tw/ref/meadows.pdf https://wtf.tw/ref/meadows.pdf
- Luc 8y ago
  Thanks, that looks very interesting!

spapas82 8y ago

We used the Boyce and DiPrima book back when I was an electrical engineering student (Elementary Differential Equations and Boundary Value Problems https://www.amazon.com/Elementary-Differential-Equations-Boundary-Problems/dp/0471089559 https://www.amazon.com/Elementary-Differential-Equations-Bou...). I remember it was an excellent book with many great examples and correlation with physics topics like mechanics, waves etc. Too bad our other mathematics books weren't at this high standard :/ Also, I really don't think that such topics can be self-studied. It seems really difficult to me to understand such topics without a teacher!

jamescostian 8y ago
> I really don't think that such topics can be self-studied Can you explain this? What makes something extremely difficult to be studied without a teacher? (I took calc in high school and never took diff eq, so my knowledge in this specific domain is basically zero)
- 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 ago
    Just 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.

gmiller123456 8y ago

While the linked course seems a lot more thorough, I took the Udacity "Differential Equations in Action" [1] course, which I found very well done. For the homework you write Python programs to compute things like gravitational slingshots, modeling epidemics, wildfires, and the n-body problem. [1] https://www.udacity.com/course/differential-equations-in-action--cs222 https://www.udacity.com/course/differential-equations-in-act...

sampo 8y ago
While the linked course notes (I looked at the pdf, definitely good textbook quality and not just course notes, really) are absolutely more thorough in the classical undergrad diff.eq. topics, the Udacity course touches some further topics (stiffness, implicit methods, control theory) that the course notes don't.
nraynaud 8y ago
Can we get to the text and videos without all the school-ish crud? (starting a course, timelines etc.)
- gmiller123456 8y ago
  Yes [1]. I haven't taken a Udacity course in quite a while, but at the time all of their videos were hosted on Youtube, so you should be able to find a lot of their courses just on Youtube. For the homework, I remember there was an include file that they didn't provide the source code for, but a Google search turned up a version someone wrote for using it locally. Usually it's pretty obvious when you've solved the problem and have the correct code, but to make absolutely sure, you have to upload your code to their server for verification. But you can take the whole class without ever visiting Udacity. [1] https://www.youtube.com/watch?v=8cLXVG2Q6D4&list=PLAwxTw4SYaPlT4WYwf_SzNA9FV0qpGE5A https://www.youtube.com/watch?v=8cLXVG2Q6D4&list=PLAwxTw4SYa...
  - nraynaud 8y ago
    thanks!

alexibm 8y ago

Look at Arnold's book on Diff, it is harder then others, but has absolutely different outlook.

selimthegrim 8y ago
That book is totally mindblowing, and obliterates artificial boundaries between physics and mathematics. It also (in the older Dover editions) had a cover where the phase portrait on the front looked like two angry eyes glaring at you that you hadn't learned enough math yet.
- Koshkin 8y ago
  Well, modern theoretical physics seems to be nothing but mathematics (mostly advanced differential geometry and group theory). And this is a good thing, as it is the sign of how far along the subject has gone in its evolution. The theory of differential equations also reduces to differential geometry, which is what Arnold’s book seems to be striving to show, as does his other excellent text on mathematical principles of classical mechanics.
ced 8y ago
I can only find the first edition on Amazon, but there's apparently a third edition? http://www.matem.unam.mx/ernesto/EDO/Ar-EDO.pdf http://www.matem.unam.mx/ernesto/EDO/Ar-EDO.pdf https://www.amazon.com/Ordinary-Differential-Equations-MIT-Press/dp/0262510189/ref=sr_1_sc_1?s=books&ie=UTF8&qid=1536220865&sr=1-1-spell&keywords=Ordinary+Differential+Equations+arnol https://www.amazon.com/Ordinary-Differential-Equations-MIT-P...
- gjm11 8y ago
  I think this is the third edition: https://www.amazon.com/x/dp/3540345639 https://www.amazon.com/x/dp/3540345639

bgorman 8y ago

When I was in college I supplemented my learning with Notes on Diffyqs, which I thought explained things much clearer and concisely than my assigned textbook. It is available for free and has tools for modeling systems in Octave/Matlab. https://www.jirka.org/diffyqs/ https://www.jirka.org/diffyqs/

TomMckenny 8y ago

On a related note, how do you take notes for math subjects with their multiline integrals and sumation and subscripts, division etc? I'm very much attached to recording every thing in simple text editors. Is there a "Notepad" or "TextEdit" for mathematical notation?

rhaps0dy 8y ago
you can use LaTeX notation and then render it with something. For example MathJax, or emacs's org-preview-latex-toggle to show the rendered equations in the same buffer you write in.
schitzapplebits 8y ago
Don’t. Pay attention, write down the things that aren’t in the text book. And then learn to read the text book. It’s a hugely valuable skill.
- thanatropism 8y ago
  I'm easily distracted, particularly from internal digressions. Taking notes helps me hold on to the thread.
takk309 8y ago
I could not imagine trying to take notes with a computer in a math heavy subject. I had to use paper for everything during my undergrad in Civil Engineering. However, if you already have a computer with you, record the lecture and review it later.
- mclehman 8y ago
  Just as a partial counterpoint to this and overall agreement with everyone else who responded, I've had success taking realtime notes on my laptop in a math class. It took a bit of practice and I'm sure I can't do it anymore, but it's not insurmountable by any means. I used latex and made liberal use of keyboard and software macros to do it, and one of the tricks was to realize that if I needed a quick-to-type way to typeset new thing X, I should just pretend I had such an implementation and make up its command on the spot. At my leisure, I could write up a conforming latex command that worked with all the notes I'd taken in realtime. That said, I've since come to realize that math notes don't help me as much as they seem to help others. I have greater success primarily listening during class and leaning on the textbook as well as online resources outside of class. I do second the use of emacs to handle the latex, but I don't think that realtime rendering is particularly important in a notes setting.

miles_matthias 8y ago

This was my favorite course in college. Super beneficial materials. On a side note, one of the reasons I loved this course was because it was online and only had 2 tests with no other assignments. The professor allowed you to schedule office hours any time you needed, but the course setup was sweet for self-studiers like me. Here's the book, Chapters 1-10 are on the midterm, Chapters 10-20 are on the final. No homework busy work, no other tests. Just 2 exams. Go.

mindcrime 8y ago

Huh... didn't expect to find a UNC-Wilmington link on HN today. I vaguely remember Dr. Herman. I think we used one of his other books in a class, or maybe he taught one of my classes. This brings back memories.

sizzzzlerz 8y ago

I took DiffEq as an EE undergrad and, to my surprise, aced the course. For some reason, the concepts just resonated with me unlike other math courses where I had to work to gain an understanding of the material. A few years later, after graduating, I was taking courses leading to an MSEE and Partial DiffEq was offered as an option that I took. Thinking it was going to be similar to regular DE, I was soon in way over my head. Totally different material and concepts. I did end up passing the course but I can honestly state that I don't think I deserved it.

tzs 8y ago

Speaking of differential equations, an oldie (1997) but goodie, "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota: https://web.williams.edu/Mathematics/lg5/Rota.pdf https://web.williams.edu/Mathematics/lg5/Rota.pdf

gp7 8y ago
> 4. Teach Changes of Variables As someone who has basically no formal training in mathematics outside what's required for undergraduate computer science this is a big one. The first time I saw it I was blown away. Everyone who knows of it seems to treat it as natural as breathing, and not worth the exposition

deleted 8y ago

[deleted]

fizixer 8y ago

The usage of terms 'initial value' and 'boundary value' is a massive failure of mathematics education. 'Initial' refers to time-like variables, and 'boundary' refers to space-like variables. There is no notion of time in mathematics. There is only a notion of space, due to geometry. I had the longest time coming to grips with the question, "mathematically what is the difference between initial value and boundary value?" only to realize the distinction is meaningless in mathematics. It's a relic of the past when differential equations were studied under physics, where time and space are a huge part of the conceptual foundation. Sometimes I wonder how much progress we would make in education if we didn't confuse the heck of our students in the name of convention and historical baggage.

ASipos 8y ago
Since time is 1D while space may have more dimensions, initial values are connected to ODEs while boundary ones with PDEs. Their behavior is quite different.