Math.321.001 Differential Equations Fall 2016
Loosely speaking, the goals of the
course are to learn the following:
Admittedly, parts of this list are vague. Paradoxically, in
order to understand a more specific set of objectives, you would
already need to know differential equations. The objectives would
mention new concepts and use terminology that you will learn in the
course, but which would not make any sense to you now (unless you have
already studied the subject).
- the important concepts, terminology, notation, and facts of
- certain types of operations, and their application to solve
- qualitative analysis methods for differential equations
- to write coherently and correctly about differential equation
models, methods, and analyses
- how and why differential equation models are used in
To make an analogy, compare this statement of course objectives with
one you might have read when first studying calculus. For that
course, a detailed and specific set of objectives would have mentioned
limits, derivatives, integrals, rules of differentiation, and so
on. But until you have studied calculus the first time, these
terms would have no meaning. Just so for the present course, a
specific and detailed list of the important concepts, techniques, and
so on would have little meaning.
So, to flesh out the objectives, here is a little overview of the
topic. In an algebraic equation, there are unknown quantities
(variables) and one of the main goals is to solve the equations to
discover the values of the variables. An answer is typically a
number or a set of numbers. In a differential equation, the
unknown is a function, and a
main goal is to solve the equation in order to find the function. In the
simplest of cases, that means finding an equation for the
Here is an example: y '
= y is a differential
equation. It concerns an unknown function, and says that the
derivative of the unknown function is actually equal to the function
itself. As you no doubt recall from calculus, ex is such a
function. So one possible solution of the differential equation
is the function y = ex.
In applications of differential equations, the unknown function often
represents how some variable of interest evolves over time. For
example, the variable might represent temperature at a specific
location (say the north pole). We recognize that this is a
function of time, and can write an equation like T = f (t) to represent that fact.
But that is not the same as knowing what the function IS. With a
thermometer we can record values of f (t) as time evolves.
What is more difficult is to predict this evolution accurately in
advance. In that sense, f (t) is an unknown function
for values of time t in the
future. Finding the unknown function means being able to predict
(ie, compute) its values in advance. If we are very lucky we can
find a differential equation that models the evolution of T accurately, and then solve
the differential equation to find the unknown function and so predict
how temperature will vary in the future.
While there are many cases in which it is possible to solve a
equation and express the answer with an equation, there are
also many equations for which this is not possible. But there are
other kinds of information that we can figure out. Does the
solution increase forever? Does it level off and approach a
steady state? Does it oscillate with a regular pattern of
repetition? Answering questions of this sort is what is meant
above by qualitative analysis,
a major emphasis of this course.
You may find qualitative
analysis more demanding than the techniques
you studied in calculus. You will have to combine information
from several different analysis tools to obtain as complete a picture
as possible of how solutions to differential equations behave. On
the other hand, many students find this sort of analysis much more
meaningful than solving calculus problems, and so more satisfying as
I hope that helps you understand the list of obectives at the
top. Please also read Is
this the right differential equations course for you?
For another point of view on the approach taken in our course, look at A Little Philosophy and a Warning on
by Robert Devaney, one of the authors of our text.