TEN LESSONS I WISH I HAD LEARNED BEFORE I STARTED TEACHING
DIFFERENTIAL EQUATIONS
GIAN-CARLO ROTA
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. The more I teach differential
equations, the less I understand the mystery of differential equations.
One of several unpleasant consequences of writing such a textbook is my being called upon
to teach the sophomore differential equations course at MIT. This course is justly viewed as the
most unpleasant undergraduate course in mathematics, by both teachers and students. Some of
my colleagues have publicly announced that they would rather resign from MIT than lecture in
sophomore differential equations. No such threat is available to me, since I am incorrectly labeled
as the one member of the department who is supposed to have some expertise in the subject, guilty
of writing an elementary textbook still in print.
The Administrative Director of the MIT mathematics department, who exercises supreme au-
thority upon the faculty’s teaching, has only to wave a copy of my book at me, while staring at me
in silence. At her prompting, I bow and fall into line; I will be the lecturer in the dreaded course
for one more year, and I will repeat the mistakes I have been making every year since I first taught
differential equations in 1958.
It is hopeless to expect that I will correct any of my mistakes at this stage of life. To allay my
feelings of guilt, I will resort to a ruse. I will present them to you in the attractive literary form
of the decalogue. The goofs, gaffes, misunderstandings, and prejudices I am about to list are not
exactly hot off the press, and you may find them cloyingly familiar. Why, then, make a public
spectacle of them? Well, I myself always find it gratifying to listen to opinions I agree with, and I
surmise that you may feel likewise as you listen to my tirade.
1. MOST OF THE MATERIAL NOW TAUGHT IN AN INTRODUCTORY DIFFERENTIAL EQUATIONS
COURSE IS HOPELESSLY OBSOLETE
Some time ago, I received a review copy of Cauchy’s introductory course in differential equa-
tions, reprinted by Springer on the anniversary of Cauchy’s death. Cauchy taught his course in the
middle of the nineteenth century, and his lecture notes were written in the attractive, flowing style
in which mathematicians of his time used to write.
It was a pleasure to read familiar topics written up by one of the great mathematicians of the
past century. But it was also a surprise to discover how little the content of the course has changed
since Cauchy. Practically the only change has been the introduction of systems, which have made
their way down the ladder since my days as a graduate student.
As I read Cauchy’s textbook, I realized how much of the material we now teach is obsolete.
The order of presentation of the outworn topics has not been altered. The most preposterous items
are found at the beginning, when the text (any text) will list a number of disconnected tricks that
Delivered at the meeting of the MAA at Simmons College, April 24, 1997.
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