# Differential Equations

By Bobby Neal Winters

These
days I spent a lot of time thinking about teaching, about learning,
about setting up systems wherein the first will facilitate the second.
We want to teach our students certain skills and certain content, but
there are other things, things more mysterious that we want to happen
too. One of these things is called knowledge integration.

When
I was in college in the early 1980s, there was a group of us who were
being educated in the sciences. This was on the down-side of the wave
that was caused by Sputnik and the Cold War. I still think of that era
as the good old days. Sure we were worried about nuclear annihilation,
but we were working.

There
was a group of use who were all taking the same classes. We would go
from computer programming to calculus and from calculus to physics.
Occasionally there were those in the group who were more experienced
and worldly who would give the rest of us the low-down on how the world
worked. It was much like learning about sex. There was the official
story that the grown-ups gave us, about doing things responsibly and
preparing for the future, that didn’t sound all that exciting at all,
but there is the unofficial story from our near-peers which is grittier
and somehow more attractive as it is full of all sorts of shortcuts and
inside information.

For
one thing, if you were in engineering you wanted to be in civil
engineering if the democrats were in, aeronautical engineering if the
republicans were, and electrical engineering if you weren’t sure, but
that electrical engineering was hard. They would also talk about the
hard classes, the ones that you should put off as soon as possible:
COBOL, Organic Chemistry, and Differential Equations.

These
classes were so hard that you begin to hear about them from your
college-age friends while you were still in high school. They were the
Unholy Trinity of the Sciences. Of these, I took Differential Equations
and I took it the first semester of my Sophomore year. I’d had every
intention of putting it off, honest, but Ken Brady, who was the Acting
Chair of the Department of Mathematics when I started college wouldn’t
hear of that. I needed to get it in as early as possible so I could go
on to take more challenging courses.

I
will grant my near-peers one thing. Differential Equations was one of
the most challenging courses I’d had in my life up to that point in
spite of having been very well prepared for it. Its one and only thing
in common with sex is that you can never understand the experience until
you’ve been through it. Indeed, this is even more so in the case of
Differential Equations as nature has prepared us for sex in a way that
it hasn’t Differential Equations. This having been said, let me try to
explain it in non-technical terms.

You
start mathematics with algebra. Then you take trigonometry which uses
algebra. Then you go into Calculus which in those days was divided into
Calculus I and Calculus II. In calculus, you do use some algebra and
some trigonometry. There will be sections here and there where you as a
student are required to recall some algebraic trivia or some arcane
formulas from trigonometry, but those instances are fairly well
quarantined from each other. There is breathing room around them. There
is time to sit back and say, “Yep, that was kind of hard, but I lived
through it.”

Differential
Equations is different. To begin with, there is the tacit assumption
on part of the teacher that you remember with perfect precision every
mathematical activity you’ve ever participated in in your entire life.
You know how to solve every polynomial equations; you know how to
evaluate every obscure integral; you are comfortable, nay, accomplished
with the arithmetic of complex numbers.

I’ve
since had the opportunity to teach this course, and I stand amazed at
the amount of work that my teacher, Mr. Phillip Briggs, was able to get
out of us. The man didn’t have a doctorate, but that didn’t matter. He
had the knack of getting us to work. You may remember the character
Fezziwig from Charles Dickens’ A Christmas Carol.
Scrooge tells the Ghost of Christmas Past, “He has the power to make
us happy or unhappy.” Well, Mr. Briggs had the power to make us work
our backsides off.

In
Differential Equations, you are learning some new concepts, but those
new concepts require that you remember some old ones. In algebra, we
learn about solving polynomial equations and obtaining their roots. We
also learn about exponential functions. In Differential Equations there
is a technique where in you use both of those things, plus keep track
of some completely new and arcane rules at the same time. Then they
throw complex numbers into the mix just for good measure.

Then
there is the amount of work involved. In algebra, most problems can be
solved in a few lines. In calculus, most can be completed in half a
page. In Differential Equations, especially when you start using
infinite series, the solution of one problem can literally go on for
several pages, and on any line of those several pages, your solution
might easily go awry. The text we used had the answer to every problem,
so that when you were done with your several pages of work you could
check to see in you were right. If you weren’t--which did happen with
astonishing frequency--you had to start all over. Which I did, even
though--and this was part of Mr. Briggs’ genius--the teacher never took
up homework!

I
knew there was something special happening at the time, but I didn’t
know the name for it and didn’t learn for many years later. I was
integrating my knowledge. We all were. We were taking things that we
had learned in separate, isolated settings and bringing them together in
a new setting. In applying our algebra in a new setting, we were
making it a part of a larger world.

To
be fair, this was happening in lesser degrees in other courses like
physics where we applied math to physical problems, but that didn’t use
such a broad variety of mathematics and didn’t use it so intensely.

Differential Equations served as a crucible for the Knowledge Integration, but the work that Mr. Briggs got out of use was the sine qua non. Like Jewel said, “There ain’t nothing for free.”

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