
Introduction to Differential Equations (2008) - peter_d_sherman
http://tutorial.math.lamar.edu/Classes/DE/Definitions.aspx
======
acjohnson55
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,
but the concepts made far more sense with a real application.

~~~
freetime2
"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 that they taught us to solve
them.

10\. TEACH CONCEPTS, NOT TRICKS

What can we expect students to get out of an elementary course in differential
equations? I reject the “bag of tricks” answer to this question. A course
taught as a bag of tricks is devoid of educational value. One year later, the
students will forget the tricks, most of which are useless anyway. The bag of
tricks mentality is, in my opinion, a defeatist mentality, and the
justifications I have heard of it, citing poor preparation of the students,
their unwillingness to learn, and the possibility of assigning clever problem
sets, are lazy ways out.

In an elementary course in differential equations, students should learn a few
basic concepts that they will remember for the rest of their lives, such as
the universal occurrence of the exponential function, stability, the
relationship between trajectories and integrals of systems, phase plane
analysis, the manipulation of the Laplace transform, perhaps even the
fascinating relationship between partial fraction decompositions and
convolutions via Laplace transforms. Who cares whether the students become
skilled at working out tricky problems? What matters is their getting a
feeling for the importance of the subject, their coming out of the course with
the conviction of the inevitability of differential equations, and with
enhanced faith in the power of mathematics. These objectives are better
achieved by stretching the students’ minds to the utmost limits of cultural
breadth of which they are capable, and by pitching the material at a level
that is just a little higher than they can reach.

We are kidding ourselves if we believe that the purpose of undergraduate
teaching is the transmission of information. Information is an accidental
feature of an elementary course in differential equations; such information
can nowadays be gotten in much better ways than sitting in a classroom. A
teacher of undergraduate courses belongs in a class with P.R. men, with
entertainers, with propagandists, with preachers, with magicians, with gurus.
Such a teacher will be successful if at the end of the course every one of his
or her students feels they have taken “a good course,” even though they may
not quite be able to pin down anything specific they have learned in the
course.

[https://web.williams.edu/Mathematics/lg5/Rota.pdf](https://web.williams.edu/Mathematics/lg5/Rota.pdf)

~~~
soVeryTired
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 :)

~~~
enriquto
definitely in the spirit of Arnold

his book on ODE is otherworldly

the quote maybe comes from his pde book

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softbuilder
Tangental anecdote: Every time I see Diff EQ mentioned the first thing that
pops into my head is the number 11. That's the score of my first, last, and
only Diff EQ test. 11%.

~~~
make3
had you studied at all? most undergrad differential equations classes are
fairly mechanical in nature, you just learn to identify the type of problem,
then you follow the steps exactly as they are written in the textbook, super
little variation or freedom

~~~
btrettel
This is often more true than many believe. But in my experience few students
get so far that they recognize the patterns and can respond in the way you
describe. That might be the crux.

I remember my undergraduate mechanical vibrations class. Every exam was
basically a test of how well you could do the Laplace transform on some linear
ODEs. I memorized the most common transforms, so this became fairly
straightforward and fast for me, but it was obvious the other students were
struggling.

If I had a problem that wasn't solveable with the Laplace transform, say a
linear ODE with variable coefficients, I'd likely take longer to do the exam,
but those never appeared in the class.

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sideproject
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?

~~~
sannee
Not really Computer Science per se, but practically every engineering field
uses numerical solvers. No one reallistically solves nontrivial differential
equations by hand these days.

~~~
cultus
Not so much in engineering, but in science, approximate perturbation
techniques are still a big deal.

------
jstewartmobile
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.

~~~
crispyambulance
While numerical methods are absolutely critical in practice, analytic methods
like you learn in what people call calculus and diff-eq are _absolutely_
essential to understanding the physical world.

You can't actually _understand_ numerical methods without a fairly deep
grounding in analytical methods.

The real problem is here is a lack of context. Engineering and most science
curriculums take a "short-cut" through mathematical education. They try to
teach just enough math to get through the major coursework. As a result you
end up with students who feel it's all just one big memorization trick .

~~~
jstewartmobile
> _analytic methods ... are _absolutely_ essential to understanding the
> physical world_

How so? My experience has been that the "physical world" is where the symbolic
approach completely breaks down.

> _Engineering and most science curriculums take a "short-cut" through
> mathematical education._

Only people taking more math than scientists and engineers would be
mathematicians. A year-and-a-half course to cover the limit, tangent-at-a-
point, functions of tangent-at-a-point, area-under-the-curve, and generalizing
all of that to higher dimensions doesn't seem like much of a "short-cut" if
you ask me.

~~~
Cobord
That is only if your allowing yourself to take the numerical methods as a
black box. If you want to be sure that your method is okay, you prove error
bounds as you take limits of 0 mesh size. Otherwise it is just building tables
of black magic for what schema to use when.

~~~
jstewartmobile
That's kind of where I was going with this. Instead of burning all of that
time on symbolic differentiation, dig-down into numerical methods ASAP so
students can get a feel for all of the related "gotchas"\--of which there are
many...

edit: IMHO, many of those "gotchas" are much more interesting than the
fundamentals of calculus.

~~~
Cobord
Consider symplectic integrators. You would never come up with them or realize
the problem of energy drift if you hadn't first paid attention to the
fundamentals of the geometry and calculus underlying the problem.

This is just my favorite example, but it illustrates how understanding the
fundamentals also explains the gotchas. Just getting a feel for them through
experience is again just black magic by building up a table of what to use
when without the generalizing principle behind it.

~~~
jstewartmobile
Never said to do away with the fundamentals. Will say that most symbolic
differentiation and integration (which is a big chunk of the coursework) is
not fundamental as much as it is fruitless busywork.

Even so, I spent a year of my time--and God knows how much of other people's
money--grinding out the mathematical equivalent of crossword puzzles so I
could get my job certificate--just like every other engineer.

Use that same time to apply the fundamentals to numerical methods, and you get
to go in far more interesting directions--like symplectic integrals, or chaos
theory.

~~~
BeetleB
>Never said to do away with the fundamentals. Will say that most symbolic
differentiation and integration (which is a big chunk of the coursework) is
not fundamental as much as it is fruitless busywork.

It's no more busywork than being able to multiply two single digit numbers in
your head. Whether it's useful to your job really depends on the job. I had a
job once in the engineering industry. When we were in meetings discussing
projects, if you could not do those types of analyses (e.g. asymptotic
behavior of certain Calc II type integrals) in your head, you would not know
what's going on. Sure, everyone could explicitly show all the steps for your
benefit, but you'd be slowing everyone down.

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Sean1708
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](http://tutorial.math.lamar.edu/Classes/DE/DE.aspx)

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

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mehrdadn
Why is this on top of HN? There are so many websites teaching introductions to
differential equations. Is there something interesting about this one in
particular?

~~~
DavidSJ
Paul’s Online Notes are a pretty well-known calculus and differential
equations reference. IMO they’re pretty decent.

~~~
zavi
Paul's Online Notes is my childhood. There's something nostalgic about seeing
this submission.

------
RivieraKid
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?

~~~
enriquto
> compare the explanation of integrating factor and exact equations

those are fringe subjects, completely irrelevant for the modern usage of
differential equations. They are useful only in computer algebra when you want
to implement differential galois theory. In practice you want to understand
the overall behavior of your system (qualitative theory) or compute particular
solutions numerically (using numerical methods, which are more precise than
evaluating the expression of the exact solution).

You'd do much better with a qualitative book about differential equations
(e.g., Arnold), about numerical analysis, or about dynamical systems (e.g.
Strogatz).

~~~
tanderson92
Integrating factors are important motivations in the design of some numerical
algorithms. Some keywords: matrix exponentials, semigroup theory, exponential
integrators.

------
budadre75
Does anyone know any website similar for advanced linear algebra and
probability with practice problems explained with detailed solutions step by
step? I know there exists other great textbook like Strang's, but I often find
textbook based learning resources lacking because of lack of detailed
solutions.

~~~
BlackFly
Not really a website, but I believe you can find ebooks online...

I highly recommend Numerical Linear Algebra by Trefethen. It gives very
detailed descriptions of particular interpretations of the singular value
decomposition and eigenvalues, it works out detailed algorithms for LU
factorization, eigenvalue/eigenvector decomposition, QR factorization etc. If
you know basic linear algebra, the book is a pleasure to read through. For
this crowd of people it is also very practical.

I don't know a good resource for probability... it is a much more diverse
subject than linear algebra (which is a very small, very detailed subset of
algebra).

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daze42
I took Cal 1 and 2 directly from Dr. Dawkins back in 2010-2011. He was the
best teacher I had in college hands down. Totally changed the way I thought
about math and I've been in love with it ever since.

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icedchai
DiffEqs... the only college class I ever got a C in!

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scrooched_moose
Oh wow, this site was an absolute godsend in college. I was stuck in a DiffEq
class with an abysmal professor and barely hanging on. I found this a few
weeks in, stopped bothering with useless lectures, and went from low-60s to
high-80s over the rest of the semester.

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jhallenworld
First and second order diffeqs with constant coefficients are a big deal, you
should at least learn that. You can solve them algebraically via Laplace
Transforms, the theory is very beautiful.

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bigasscoffee
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.

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

~~~
ballooney
This is why people suggest that terms like 'software carpentry' more
accurately capture what 99.5% of programmers do than 'software engineering'.

