Math.321.001 Differential Equations Fall 2016
Is this the right course for you?
As explained in the page on Course Objectives, this course will
emphasize qualitative analysis of differential equations. This is
a fairly modern approach to the topic, and will differ from traditional
calculus courses in important ways. Please read further for more
details about the qualitative approach and why I prefer it. After
reading the rest of the page, if you think you would prefer a
differential equations course with a more traditional emphasis, please
consider transferring to the other section, meeting 9:45 - 11 AM MTh.
In describing the qualitative approach, I will quote from this webpage
posted by Robert Devaney, one of the authors of our textbook. He
is a course in ordinary differential equations. However, the course is
by no means "ordinary." In years past, before we had widespread access
to computers and computer graphics,
courses in ordinary differential equations consisted mainly in a
series of special tricks to solve special differential equations.
The tricks Devaney is referring to here are similar to methods of
integration studied in calc 2. But the need for students to
master a large tnumber
of techniques in either of these areas has greatly diminished since the
advent of computer algebra systems (such as Mathematica or WolframAlpha), that can apply
the techniques faster and more accurately than a human working problems
out by hand. If an application involves differential
equations that can be solved using traditional methods, it is usually
better to rely on software. Devaney says a modern course in
differential equations should focus instead on differential equations
that can NOT be solved by the traditional methods. Continuing to
most differential equations (and in particular most
differential equations that arise in applications) cannot be solved
explicitly by these or any other method.
What does solving explicitly mean?
It refers to finding an equation for a function f (t)
that satisfies the given differential equation. To use an
analogy, the equation ex
= x + 5 cannot be solved using algebra. There is no way
to combine the familiar operations and functions of calculus to obtain
an exact solution. By trial and error (or more
sophisticated methods) we can find an approximate solution is given by x
= -4.99321618864790.... Something similar occurs
with differential equations, but in that context the unknown is a
function, not a number. For example, consider the differential
equation y' y = 2et. Finding a solution
means finding a function for which multiplying the function by its own
derivative results in 2et. One possibility is y = 2et/2. That is an explicit
solution. In contrast, the differential equation y' y = 2e ^ t2 (that is e with an exponent of t2 ) does
not have an explicit solution: there is no familiar calculus function
that can be substituted for y
to make the equation true.
we all have access to high-speed computers and computer graphics. Like
humans, computers cannot solve most differential equations that arise.
However, they can give us an APPROXIMATE or NUMERICAL solution. For
many purposes, this is good enough.
The last sentence is key. Devaney is a nationally recognized
expert on differential equations and applications. He is telling
us here that the traditional differential equations course will be more
comfortable to many students, but may not be terribly relevant to
actual applications students will encounter in their
careers. Of course, some students are not motivated by
applications, and enjoy learning about math for its own sake.
From that perspective, there is a lot of interesting material in a
traditional differential equations course. My point here is not
to hold up one approach or the other as better. Rather, I want
students to be aware of the different approaches that can be taken, and
to make an informed decision about which course best suits their needs
Unfortunately, computers make
mistakes (sometimes because of round-off errors or sometimes because
the differential equation is not suited to numerical approximation). So
we always have to be careful when we solve differential equations this
way. More importantly, the output of the computer is not a formula that
we can use to compute values of our solutions. Rather, the output from
the computer is a rather lengthy list of numbers. Most often, it is
best to view this list geometrically as a phase line or plane or other
All of this means that this will be a very different type of
mathematics course. In this course you will rarely be asked to generate
specific formulas for solutions of differential equations. Rather, you
will be asked to understand the algorithms that lead to numerical
solutions, to interpret the resulting pictures produced by the
computer, and to relate these images back to the original application.
.... Your homework problems and questions on exams will often involve
essays rather than simple routine computations. And you will often have
to use technology to come up with answers to questions that are posed.
... Most students in the past have found this kind of course quite
challenging, but lots of fun. If you are used to the old style of
mathematics courses, be prepared for something quite different and
perhaps much more relevant to whatever your use for differential