
Ten lessons I wish I had learned before teaching differential equations (1997) [pdf] - JMStewy
http://www.math.toronto.edu/lgoldmak/Rota.pdf
======
mightybyte
I couldn't agree with the last point more.

> 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...In an elementary course in differential equations, students
> should learn a few basic concepts that they will remember for the rest of
> their lives...

I hated the DE cleass I took in college and it was largely because I felt like
it was nothing but a bag of tricks. I very distinctly remember one problem
that seemed unsolvable until the teacher showed that you had to substitute a
"2" with "1/2 + 3/2". And then, to make matters worse, he put the exact same
problem on the test. So we were being rewarded, not for really understanding
the core basic concepts, but for memorizing the tricks needed to solve
specific problems.

~~~
batbomb
I'm not sure what that problem would have been, but an integral part (no pun
intended) of DEs is expansions. It's not part of a bag of tricks, it's a very
common technique used to solve a problem.

~~~
vcarl
I remember a similar problem in Calc 2. I forget the specifics now, but I
think it was an integral of some combination of sin/cos that ended up being
circular. You had to recognize an opportunity to swap one of the steps for an
equivalent, which would lead you to the final solution.

~~~
mattb314
Probably the second example here [1] for those curious (I think the integral
of sin(x)*e^x dx is the only place I've seen this used, would love to know if
there are other examples).

[1]
[https://en.m.wikipedia.org/wiki/Integration_by_parts#Tabular...](https://en.m.wikipedia.org/wiki/Integration_by_parts#Tabular_integration_by_parts)

~~~
octonion
This becomes much more transparent if you realize you're integrating Im(e^x *
e^{ix}). And it's no longer a trick but a technique.

~~~
vcarl
It would be great if that were how it was taught, but when I took the class it
was taught as a trick. No theory behind it, just the prof on the board saying,
"But look! :swap: And now you can integrate it."

------
graycat
In the OP, the author Gian-Carlo Rota started out with:

> One of many mistakes of my youth was writing a textbook in ordinary
> differential equations. It set me back several years in my career in
> mathematics. However, it had a redeeming feature: it led me to realize that
> I had no idea what a differential equation is.

Wow! Good to see that he wrote this. Looking at his book,

Garrett Birkhoff and Gian-Carlo Rota, _Ordinary Differential Equations_ , Ginn
and Company, Boston, 1962.

I got the same impression! I couldn't see what the heck they were driving at.
Instead, they seemed to flit around with a lot of tiny topics of little or no
interest for little or no reason.

Want to understand ordinary differential equations, read Coddington:

Earl A. Coddington, _An Introduction to Ordinary Differential Equations_ ,
Prentice-Hall, Englewood Cliffs, NJ, 1961.

Then for more, to make such equations much more important, read some
deterministic optimal control theory, e.g., Athans and Falb

Michael Athans and Peter L. Falb, _Optimal Control: An Introduction to the
Theory and Its Applications_ , McGraw-Hill Book Company, New York,

that, BTW, also has some good introductory, but very useful, material on
ordinary differential equations.

More generally, want to know what to study in a subject that will be useful?
Okay, one approach is to go to more advanced material that is an application
of that subject and see what that material emphasizes for prerequisites, e.g.,
sometimes quite clear in an appendix.

E.g., Athans and Falb say quite clearly what is important in ordinary
differential equations for their work.

------
mynegation
My major is computational math, from 15 years ago, from leading Russian
university, so it is just anecdata, and by no means should be generalized.

I absolutely love mathematics, for me it is the embodiment of pure beauty.
Still, I positively, absolutely hated the sophomore course of ODEs. The way it
was taught was extremely abstract: here is the equation, this is integration,
this is separation, this is your SLP, now go deal with it.

It was totally pointless and life-sucking. It was not until I got to the 3rd
year and learned about specific applications in physics (like heat
dissipation, strings, and springs), and later in finance (stochastic calculus)
and biology (e.g. Lotka-Volterra) when I realized how many wonderful and
extremely useful applications they have.

Have this course started with that, things would be completely different.

~~~
my5thaccount
I took, we called it Diff-e-q, my freshman year in college and the professor
died a couple weeks into the course and the new guy was not pleased about
having to teach it. We did the applied aspects with the springs and the little
ants running away from a candle heating the corner of a plate.

What I didn't like about it was just all the memorization. I had no desire to
memorize a bunch of formulas that I knew full well in the real world I'd look
up in a table or type into a computer. What I wanted to learn how to do was
solve problems using math, not memorize patterns of formulas to apply to
problems.

So I didn't memorize them and instead went to work and earned money to pay for
college. Still passed the class but it was one of my lowest grades. It's a
hard class even without the memorization.

~~~
vlasev
It's a little spooky that the author, Gian-Carlo Rota, passed away the night
before teaching a class.

~~~
my5thaccount
That is spooky. Both from heart problems too. Maybe the stress of teaching the
class takes a toll. Heart attacks are pretty common among men of that age,
though.

Coincidence, of course. Spooky. For sure.

------
tomekowal
Please, write another textbook.

The internet changed drastically the process of writing books. I saw people
making profit from books available online for free. I saw books written
chapter by chapter with errors found quickly by first readers. I saw systems
that allow commenting parts that are not clear enough with comments how to
clarify them.

If you promise to deliver a textbook that teaches skills relevant to
engineers, they will fund the time it takes to write the book. I would spare
couple of dollars even if you said that it will take 5 years. I believe, there
will be even companies that will give you funds upfront. If you reach out for
help there will be people who will help you collect example problems from
different fields to replace couple of "salt tank problems".

I am not a mathematician, so I don't know how mathematics textbooks are
written and how much effort goes into them, so feel free to point out that
this idea is stupid.

~~~
Terribledactyl
Rota died almost 17 years ago, I'm sorry.

------
p4wnc6
I'm proud to report that at my undergrad institution, Rose-Hulman Institute of
Technology, they very successfully adhered to these rules (and I was taking
ODEs there way back in 2005).

They had a custom textbook created for their 2-course ODE sequence that
several of the faculty collaborated on. Though it did contain content on
uniqueness theorems and some proofs, far and away the biggest two items
hammered in were (a) linear equations with constant coefficients, and (b)
Laplace transform methods.

They also offered (at the time) a 3rd, optional course called Boundary Value
Problems that was focused on several physics-motivated BVPs like with
Laplace's equation, heat equation, wave equation, Young's modulus, and others,
and that course heavily used Fourier and Laplace methods.

We did have word problems, but they were almost exclusively "salt tank"
problems. Literally, every word problem described a tank of water or pre-mixed
brine solution, with some description of either more salt or more water being
added or removed, either gradually or in discrete injections.

The fact that _every_ problem was an infamous "salt-tank problem" essentially
made its status as a word problem irrelevant. This seems like it wouldn't be
that helpful but actually it was really nice. You got so used to the different
pieces that comprised the modeling problem that when you went off and did
something in other courses, like circuit systems or conservation systems in
mechanical engineering, you knew how to translate the problem to 'salt tank'
form, which really covered a huge range of practical problems.

As a math major, one fault I noticed of this method was that it did not make
the connections to linear algebra very clear. It took me another few semesters
afterward to catch up on that part, but I can understand how engineering
majors cared less about that.

I don't know what Rose-Hulman does for this curriculum now, but it would be
cool to somehow take a "snapshot" of their methods for it and compare it with
other experiences like this OP.

~~~
tnecniv
> It took me another few semesters afterward to catch up on that part, but I
> can understand how engineering majors cared less about that.

It's definitely something engineers care about, but not at the undergraduate
level.

------
analog31
I once told a math teacher at a Big Ten university, that I thought their
undergrad math instruction for engineers was weak. As an example, I said that
I didn't think students learned any engineering applications of differential
equations. He looked at me with a straight face and said: "There are no
engineering applications of differential equations."

~~~
zerohp
My undergrad was at a Big Ten university. The analog signal processing course
in EE teaches how to use differential equations to solve circuits. Right after
they taught it, we learned laplace transform and never looked at a
differential equation again.

~~~
tnecniv
Same with my EE undergrad. Then I got to grad school and decided to take a
course in Linear Systems, which is when I realized my ODEs course taught me
nothing.

------
drjesusphd
> I do not know how to properly motivate the Laplace transform

I feel like this is impossible without going to the complex plane. Like the
author said, taking the inverse Laplace transform is no joke.

I feel like I never properly understood the Laplace transform until I learned
about Landau damping. This is when waves exist, but are damped in a
collisionless plasma. This damping is _not_ disspiation and the energy does
not get converted into heat. The usual way of presenting this is to show that
if one _Fourier_ transforms in time, you get the wrong answer. The fact that
the system begins at a certain state, and is thus an initial value problem,
needs to be respected.

~~~
semi-extrinsic
The Laplace transform is somehow the continuous analog of a Taylor series
expansion. You don't need complex analysis to motivate it. I sketched this for
my students when TAing once upon a time, heavily inspired by this [1] nice MIT
lecture.

I'm very surprised this isn't standard material. It makes the parallel between
Laplace and Fourier transforms so much more intuitive, because you get Taylor
series as a parallel to Fourier series.

[1] [https://youtu.be/zvbdoSeGAgI](https://youtu.be/zvbdoSeGAgI)

~~~
nimish
expand an analytic function in its taylor series then find its values on the
unit circle. There's a Fourier series.

But the Fourier series uses global data than the taylor series which uses
point data so they aren't perfect analogs.

A laplace transform is a fourier transform rotated in the complex plane (more
or less), and if you allow the transform to take complex "frequencies" then
they are basically unified. The difference is that the laplace transform is
all about causal functions of time (t<0 => f(t) = 0) where as the fourier
transform is less picky.

------
Kinnard
"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"

~~~
graycat
Yes, and once students discover this, then good luck in getting students in
the class!

And, if regard the material on differential equations as essentially nonsense,
then good luck getting NSF grants for research in the subject!

Actually, can communicate a lot of good information in a course in
differential equations, but to do this apparently need some exposure to some
of the leading applications of differential equations.

~~~
GFK_of_xmaspast
> And, if regard the material on differential equations as essentially
> nonsense, then good luck getting NSF grants for research in the subject

Luckily, in this world, NSF grant writers are not typically undergraduate
students, and there's plenty of money flowing towards ODE research (here are
some examples:
[http://www.nsf.gov/awardsearch/showAward?AWD_ID=1600381&Hist...](http://www.nsf.gov/awardsearch/showAward?AWD_ID=1600381&HistoricalAwards=false)
[http://www.nsf.gov/awardsearch/showAward?AWD_ID=1408295&Hist...](http://www.nsf.gov/awardsearch/showAward?AWD_ID=1408295&HistoricalAwards=false)
[http://www.nsf.gov/awardsearch/showAward?AWD_ID=1318480&Hist...](http://www.nsf.gov/awardsearch/showAward?AWD_ID=1318480&HistoricalAwards=false)
[http://www.nsf.gov/awardsearch/showAward?AWD_ID=1505215&Hist...](http://www.nsf.gov/awardsearch/showAward?AWD_ID=1505215&HistoricalAwards=false)
[http://www.nsf.gov/awardsearch/showAward?AWD_ID=1418042&Hist...](http://www.nsf.gov/awardsearch/showAward?AWD_ID=1418042&HistoricalAwards=false)
[http://www.nsf.gov/awardsearch/showAward?AWD_ID=1346876&Hist...](http://www.nsf.gov/awardsearch/showAward?AWD_ID=1346876&HistoricalAwards=false))
and of course there is much more cash going into PDEs.

~~~
graycat
Nice list.

I saw emphasis on control, optimization, numerical methods, and applications.
Nice.

And, sure, PDE's stand to get more.

------
backprojection
> Some thirty or so years ago, Bessel functions were included in the syllabus,
> but in our day they are out of the question. > Teaching a subject of which
> no honest examples can be given is, in my opinion, demoralizing.

I don't get this. Differential equations theory is about proving existence and
uniqueness of solutions. If you have to use numerical techniques to actually
compute the solution, then that's perfectly fine. After all, even if the
solution is explicit, like sin(x), or especially a special function, then we
still need to use numerical techniques to actually evaluate that explicit
solution.

~~~
CamperBob2
I don't get it, either. Bessel functions certainly do have engineering
applications.

~~~
zevets
I recently saw them in a grad eng class, but I agree with the article - from
what I saw there is no need to give them the math professor treatment. You can
use them as a piece of trivia - ie pde of type x has this set of basis
functions - now apply the principles of basis functions to solve your your
problem.

------
pkrumins
One lesson academics should learn: pdf-naming-skills.pdf.

I've been collecting interesting scientific papers and publications since
early 2000 (I've a collection of 10,000 or so) and I've not yet seen a single
academic, not even a computer scientist, who understands how to name your
documents right so that when I download them I could quickly find them. I've
to rename every single pdf. It's infuriating.

Someone should teach academics an SEO course.

~~~
GFK_of_xmaspast
Everybody has a different system, and why should they standardize on yours?

~~~
pkrumins
What kind of a system is surname.pdf, New-Pub_New2new.pdf, USENIX_.PDF, and
Paper.pdf.

I like to say: Show me how you organize your files, and I'll tell you how good
of a computer user you are.

~~~
peeters
How good of a computer user do you have to be to sort by modified date? :P

~~~
echelon
Lots of things can touch modification date.

It's ironic, because professors and instructors often ask their students to
name their research papers, essays, and projects with a well-defined,
searchable naming scheme.

------
swehner
The last point may well be the best, "TEACH CONCEPTS, NOT TRICKS"

~~~
sonabinu
I am taking a refresher class in calculus II from a community college and am
pained by the tricks being taught and feel that the students are being
deprived of learning core concepts. I am taking the trouble to read up and not
too focused on the solution techniques that the professor emphasizes. However
this doesn't do too much for grades. I wish he focused on concepts and
application.

~~~
HarryHirsch
But the bag of tricks is a feature of US undergrad education seemingly
everywhere, except perhaps in pockets at the very highest level. The tricks
permit the student to pass the test so they can go on to do something else.
Whoever sits in a chemistry course isn't there for the chemistry, they are
mostly there to go up to medical school or allied health science. Of course
there are entrance exams.

We have stopped putting things into context, i.e. we do not provide an
education any longer. The sideswipe remark in the original paper about Prof.
Neanderthaler is also very real.

------
vlasev
He was the author of Indiscrete Thoughts[1], a great book on Mathematics.

[1] Review here: [http://www.maa.org/publications/maa-reviews/indiscrete-
thoug...](http://www.maa.org/publications/maa-reviews/indiscrete-thoughts-0)

------
solipsism
Could someone give some examples of applications of numerical methods for
solving differential equations that are relevant to a HN crowd? Also, where
might I find some introductory material that teaches it well, according to the
the suggestions in the OP?

~~~
cossatot
I'm not sure what the HN crowd finds useful as a group, but I personally use a
couple of different techniques in my work (geophysics). One of them is finite
difference and finite element methods. I have a book called 'Introduction to
Numerical Geodynamic Modeling' by Teras Gerya that teaches finite difference
modeling of plate tectonic phenomena, particularly of 2nd order differential
equations (Poisson equation and variants) that are useful in modeling heat
flow, diffusion and so forth through space and time. It's a great book but
written for a specialized audience. I've used finite element models a lot, and
the occasional boundary element method, but never written any.

I also use Green's functions, which are equations that describe the response
of a medium to an impulse (think the propagation of sound waves from a source,
though I do different stuff), by using convolution.

But I think jofer is the only other geophysicist on HN so we're probably not
representative. Nonetheless, a lot of HNers have a physics, classical
engineering or chemistry background and use similar tools... just not to find
out what happened tens of millions of years ago.

------
mathgenius
Does anyone else get the urge to spend the next few hours (or days) trying to
work out (at least the basics of) everything Rota is talking about here?

Mathematicians have so much fun..

------
yiyus
Although I agree that word problems can be somewhat distracting in a DE
course, I do not think they should be totally avoided. One of the main reasons
it was difficult to grasp the concepts we studied in our first DE courses is
that they were too abstract. When later we studied other subjects, such as
electric circuits or fluid dynamics, everything started making much more
sense.

In my opinion, the ideal way of learning would be to first have very basic
(only conceptual) introductory courses of applied fields, where we find some
basic equations that we do not know how to solve. And then, we study DE to
learn the techniques to solve these problems, avoiding direct references but
keeping in mind where we are going with all this.

------
musgravepeter
I thought the comment on exterior differential forms was interesting. I always
wanted to delve into those and better understand what a dx all by itself was
when separated from dy/dx by simple manipulation. Loved his comment "We
justify this sudden introduction of differentials by saying that this is 'just
another way or rewriting the differential equation,' or some equally atrocious
lie."

I am now lusting after [http://www.amazon.ca/Exterior-Analysis-Using-
Applications-Di...](http://www.amazon.ca/Exterior-Analysis-Using-Applications-
Differential/dp/0124159028) but it's a bit pricy for a indulgence purchase!

------
seansmccullough
I got an A- in my differential equations class in college. I still wasn't sure
what a differential equations was at the end. My pattern matching skills got a
good workout, though.

~~~
vlasev
I had a similar experience with similar grade and outcome. I finally started
understanding things when I took PDEs and so on.

------
rorykoehler
It seems there are lots of posts complaining about how ODE's are taught here.
I am planning to study them in the next months. I can only learn maths through
applied mathematics and always need to know the why before I can get the how.
Can anyone please point me in the right direction for online materials which
will help me self-learn, taking into account my learning preferences?

~~~
raverbashing
I would say skip studying "pure ODEs" then examine what you need specifically
depending on your area

------
antman
Can anybody comment on that? (page 8 paragraph 1 ):

 _Professional mathematicians have avoided facing up to density functions by a
variety of escapes, such as Stieltjes integrals, measures, etc. But the fact
is that the current notation for density functions in physics and engineering
is provably superior, and we had better face up to it squarely_

~~~
gjm11
In physics and engineering, you traditionally talk about things like "delta
functions", write expressions involving them as if they are actual functions,
etc. This is notationally very convenient but may be misleading because these
things are not really functions.

So, what _are_ they really? Well, the key things you can do with them are (1)
"boring" linear algebra operations (you can add and subtract them, and
multiply them by scalars) and (2) multiplying by some function and taking the
integral. E.g., what delta(x) -- the Dirac delta function -- really is, is a
thing such that when you compute integral f(x) delta(x) dx, you get f(0).

And so pure mathematicians have ways of dealing with them that make this
property more explicit. The theory of _distributions_ says: no, these aren't
functions, they're linear functionals on the space of functions (e.g., the
delta function is the thing that maps f to f(0)). So now you're no longer
allowed to write them as integrals, which means that the very close analogy
between "distributions" and ordinary functions is obscured, and e.g. if you
need to do a change of variables you can no longer just do it the same way you
already know about from doing integrals.

Alternatively, the theory of _signed measures_ says: no, these aren't
functions, they're kinda like probability distributions except that the total
"weight" doesn't need to be 1 and the density can be negative in places. They
are naturally applied not to _points_ but to _sets of points_. (E.g., the
delta function is the signed measure that gives a measure of 1 to any set
including 0 and a measure of 0 to any other set.) Now you _are_ allowed to
write those integrals, but instead of writing _integral f(x) delta(x) dx_ you
need to write _integral f(x) dH(x)_ where H(x) is the "Heaviside step
function", so instead of delta(x) appearing there you have (morally) its
integral, and again if you want to change variables or something you need to
know a new set of rules for what you do to the measure.

Note: I have skated over some technicalities. They are quite important
technicalities. Sorry about that.

The sloppy non-rigorous physicists' and engineers' notation, where you just
pretend the damn thing is a function and manipulate it as you would any other
function, is more convenient. (Right up to the point where you do some
manipulation that is safe for actual functions but gives nonsense when applied
to singular things like delta functions, and get the wrong answer.)

It's a little like calculus notation. The "Leibniz" notation we all use these
days writes derivatives as dy/dx as if dx and dy were just small numbers
(compare: we write integrals against distributions as integral f(x) delta(x)
dx as if delta were just a function), which is kinda nonsensical if you take
it too seriously but very convenient because it makes things like dz/dy dy/dx
= dz/dx "obvious", which is not just coincidence but has something to do with
the fact that derivatives really are kinda like quotients (in fact, they are
limits of quotients). Similarly, using "function" notation for distributions
lets you write things like "integral f(x) delta(x-3) dx" and see that "of
course" that's f(3), and this convenience isn't mere coincidence but has
something to do with the fact that distributions really are kinda like
functions (and in fact every distribution "is" a limit of functions).

Newton had a different notation for derivatives. It didn't have a conceptual
error baked into it (pretending that derivatives just _are_ quotients), but it
turns out that that's a _useful_ conceptual error and that's part of why
everyone uses Leibniz's notation these days.

------
al2o3cr
+1 to teaching concepts - but I don't agree with the author's opposition to
word problems and learning to apply a "bag of tricks". I'd argue that
"deciphering vaguely-phrased word problems and figuring out which of a
selection of tricks to apply" is the MOST transferrable skill somebody can
take away from a math class, because it's a major component of working in lots
of other fields. For instance, the biggest difficulties I've observed in
novice developers are in breaking apart a big challenge ("write a program that
solves this Sudoku board") into digestible / implementable pieces and in
understanding which piece of information they already know can get the result
they want.

~~~
amoonki
I agree, but I think the "word problems" the author was referring to are much
lower quality than the ones you're thinking of. I imagined some highly-
contrived exercises where all of the relevant information is already pre-
processed for you, removing any need for problem decomposition. For example,
"if the angle between the ground and a tree's shadow is 45 degrees and a 50 ft
tall telephone pole that's 10 ft away from the tree casts a shadow..."
(substitute a similar differential equation problem). As you point out, half
the fun is defining a problem and breaking it down, and these kinds of word
problems don't give you a chance to do that.

------
danidiaz
Where can I find an expansion of the intuitive explanation given for
integrating factors?

> It is of the utmost importance to explain the relation between the solutions
> of the differential equation and the solutions of the system. The solutions
> of the system are trajectories, they are parametric curves endowed with a
> velocity given by the vector field. The solutions of the corresponding
> differential equation are integral curves, and their graphs are the graphs
> of the trajectories deprived of velocity. Often, instead of solving the
> differential equation, it is more convenient to solve the corresponding
> autonomous system.

~~~
bzbarsky
Here's an attempted expansion.

If you have an equation of the form dy/dx = f(x) and you want "solve" it, what
you are typically looking for is to write y = g(x), right? In other words, the
solution to this differential equation is some curve in the x-y plane. This
applies more generally, e.g. to situations where you end up with an "implicit"
solution like h(y) = g(x): you still get an equation relating x and y which
can then be represented as some set of points in the x-y plane (the ones that
satisfy that equation).

Now say f(x) happens to have the form a(x,y)/b(x,y). You can consider the
system of two differential equations: dy/dt = a(x,y), dx/dt = b(x,y). Solving
this system gives x and y as functions of t. Picking any particular value of t
gives values of x and y, which gives you a point in the x-y plane. The first
key point is that the set of points produced by this procedure as you plug in
all possible values of t is exactly the set of points for which the h(y) =
g(x) equation above holds. In other words, the solution to the two-equation
system encapsulates all the information about the solution to the original
equation.

The second key point is that the solution to the two-equation system has
_more_ information than the solution to the original equation. In particular,
it has the actual values of dx/dt and dy/dt for every given value of t, which
don't correspond to anything in our original problem. Their _ratio_ does
correspond to something in our original problem: the slope of the tangent line
to the solution curve (dy/dx). But the exact values themselves are somewhat
arbitrary, as long as their ratio is correct. Put another way, our original
problem's solution is a curve in the x-y plane, while the solution of our two-
equation system is a curve together with a description for how fast to move
along it as t changes. That's the "velocity" bit in Rota's article.

OK, but if how fast we move along the curve doesn't really matter, maybe we
can choose to move along it in a nice way that makes it particularly simple to
figure out what the shape of the curve is. Our only constraint is that at any
given point along the curve the ratio of dx/dt and dy/dt is fixed, because in
our original problem we have a fixed dy/dx if we're given values of x and y.
So if, at every point (x,y) we multiply dx/dt and dy/dt by the same number
(which can depend on x and y) then we get a system of two equations that has
different solutions for x and y as functions of t, but the graph of the
resulting thing in the x-y plane still looks the same. That's the integrating
factor bit; we just formalize it by saying that we multiply both dx/dt and
dy/dt by the same function q(x,t), which is exactly what it means to multiply
them both at every point by some number that might depend on that point.

The hard part, of course, is choosing a q(x,y) that makes things work out
nicely and makes it easy to solve our two equations to get x(t) and y(t).

Here's a concrete example that might help:

Say dy/dx = x/y. We rewrite this in the form dy/dt = x, dx/dt = y. This isn't
terribly convenient to solve, so we multiply by q(x,y) = 1/(2xy) to get a new
system: dy/dt = 1/(2y), dx/dt = 1/(2x). At this point, maybe you just look at
it and go, ah, y = sqrt(t + C1), x = sqrt(t + C2), or maybe you figure out
some other way to get there. In any case, now you see that t + C1 = y^2, t +
C2 = x^2, so x^2 - y^2 = C for some constant (C2-C1, but both are arbitrary,
so this is just some single arbitrary constant). And that's your (implicit)
solution for the original differential equation: a hyperbola, or more
precisely a family of hyperbolas each of which satisfies the equation.

To illustrate the point about velocities, let's just consider C = 1, so x^2 -
y^2 = 1. The point (sqrt(2), 1) lies on this curve. At this point, dy/dx = x/y
= sqrt(2). On our original formulation of the parametric system, dy/dt =
sqrt(2), dx/dt = 1 at this point. In our reformulation with the integrating
factor, dy/dt = 1/2 and dx/dt = 1/(2*sqrt(2)). So the two formulations have us
moving along the hyperbola at different speeds at this point as t changes, but
they're moving along the same hyperbola.

Does that help at all?

~~~
danidiaz
It helped, thanks!

------
XaspR8d
I somehow managed to get a bachelor's in mathematics without ever taking a DE
class (snuck through on a year when program requirements were being
rewritten). Everyone I've ever told this to has been aghast, and yet when I
ask what I missed out on, no one really has a response other than "I thought
everyone had to take it".

------
RickHull
Wow. I managed to skip Diff-e-q in my C.S. curriculum, and I always wondered
if was the worse for it. Perhaps I was prescient.

------
gajomi
I would be very curious to see if Gian Carlo Rota had anything to say about
Stephen Strogatz's view on this. Strogatz's text (which was written three
years before this article, right before he left MIT), is much beloved by many
scientists and engineers, but most mathematician's will have complaints about
it.

~~~
ataspinar
Do you have a link?

~~~
JMStewy
I believe he's referring to Strogatz's introductory textbook "Nonlinear
Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and
Engineering".

Amazon link: [http://www.amazon.com/Nonlinear-Dynamics-Chaos-
Applications-...](http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-
Nonlinearity/dp/0813349109)

------
rafinha
"FORGET ABOUT EXISTENCE AND UNIQUENESS OF SOLUTIONS" what? most important
thing about differential equations.

~~~
kevinr
As an engineer, I promise you, I give not one single solitary conscious fuck
about the uniqueness of solutions to differential equations. Mostly I just hit
things with Fourier or Laplace transforms as appropriate until they stop
moving.

~~~
yiyus
As an engineer working in some optimization problems where is utterly
important if a solution is a local or a global minimum, I assure you I give
many fucks about the uniqueness of solutions.

I totally agree with your way of working for most applications, it is what I
do most of the time too, and I agree with the article in that this point is
given more importance in DE courses than it really is for engineers, but there
are perfectly valid use cases for these theorems in engineering.

~~~
kevinr
I'm glad _someone_ got something out of that section of the course besides a
bad grade and the lingering suspicion that if it had been taught out of the
engineering school it would have made more sense.

------
chris_wot
Very late to the conversation, but I emailed this to my uncle and he wrote
back:

 _At ANSTO I worked with solution of simultaneous first order differential
equations, as arose from Newtons law of cooling for a 4-body calorimeter. That
was fun._

------
madengr
I remember my undergrad signals and systems class. Instructor said if I use
Laplace transforms on any of the problems, I would get no partial credit. I
got an A+ for the course.

Also got A+ in DE, but I still don't think a grokked it.

------
jonesb6
I 100% believe DE 1 and 2 are courses used to weed out computer science
students who don't meet a certain criteria. Whether this is good or bad is
highly debatable imo.

------
dbpokorny
> As a matter of fact, the need for proving existence theorems was not felt
> until the end of the nineteenth century, and I refuse to believe that
> someone like Cauchy or Riemann did not think of them. More probably, they
> thought about the possibility of proving existence theorems, but they
> rejected it as inferior mathematics.

...

> Most often, some student will retort with the dreaded question: “So what?”

Insecure snob.

------
forkandwait
a = F/m

Want motivation? Just saying.

