LINEAR ALGEBRA AND DIFFERENTIAL EQUATIONS

Objectives: To study both the computational use of matrices and their connection
with vector spaces; to use this understanding to understand the theory of differential
equations; and to develop and apply various techniques for solving these equations.

Subject material

A differential equation is an equation involving an unknown function and its various derivatives.
You studied the simplest sort of these in calculus: “Solve ” is another
way of saying “Calculate ”.

Like integrals, differential equations are intrinsically “hard”, in the sense that there is no
systematic procedure which will produce explicit solutions to all of them. But certain types
which arise frequently in practice can be solved explicitly, and one of our major goals for the
course is to learn some of the fundamental techniques.

We will begin with a recap and more thorough examination of where MAT110 leaves
off: with a few examples of how differential equations arise in practice, how we can gain a
qualitative understanding of them using slope fields, and how we can explicitly solve some
of them using separation of variables.

We will see along the way that we frequently need to solve systems of linear equations.
We will also see that the general solution of an nth order differential equation typically has n
“degrees of freedom” associated with it. These two observations lead us to two of the central
subjects in linear algebra: Matrices encapsulate the computational side of solving equations,
and the concept of vector spaces formalizes the geometry and algebra of our “degrees of
freedom”. We will study both topics fairly extensively in their own right.

Next we will survey a few particular solution techniques for first-order DEs and some of
their applications. The linear algebra will lurk in the background for a while, until it reappears
more essentially when we finish the course with higher order systems.

I expect that we will cover most of the topics in the first six chapters of Peterson and
Sochacki’s book, with occasional digressions for perspectives (especially geometric ones)
other than theirs. The goal of learning differential equations and linear algebra in a single
course is an ambitious one, and we will move quickly. It is very important that you not fall
behind.

Assignments and grades

Assignments: If you’ve made it this far in mathematics, you know the importance of
practice. Problem sets will form a very substantial part of your grade – both in themselves,
and because they form the foundation of learning for the exams. I encourage you to discuss
the problems with me, each other, and the tutors. But you should write your final solutions
on your own, without consulting anything more than sketchy notes that you understand
thoroughly. Submitting someone else’s work as your own is plagiarism. It is unethical and
educationally counterproductive.

I will assign problem sets regularly and collect and grade your accumulated solutions every
week or so. Some exercises will be routine practice, some more theoretical, and occasionally
some may be designed to guide you in a more thorough exploration in a topic than we have
time for in the classroom or on exams. Please put your name on every page, write neatly,
label each problem, and staple your pages together in order. Leave margins with room for
comments and corrections.

Your first assignment is easy: Sometime during the first week of class (by January 28,
to be specific), come to my office and ask me a question about mathematics. It can be
anything – course work, general curiosity, long division, the theological significance of the
incompleteness theorem... – my main objective is to encourage open dialogue from
the very beginning. I’ll also ask you about your mathematical (and related) background,
and what kind of things I can do to make the course go well for you. (I know many of you
from earlier courses, and look forward to the chance to say hello again. Welcome back! )

Exams: We will have two midterm exams in class, tentatively on February 27 and April
10, and a three-hour cumulative final to be scheduled by the registrar during finals week,
May 5 – 9.

Exams will include a mix of very routine computational problems and applications, and
more abstract and conceptual ones.

Grades: You will be assigned a percentage grade based on the following weighted ingredients:

Your percentage grade will then be converted to a letter grade using an increasing function.
(Grades in the 90s will correspond roughly to the A range; 80s, the B range; and so forth,
subject to some fine tuning at the end.)

Calculators and computers. I will not require it, but you may find it useful to have a
calculator which graphs and numerically integrates functions. But be aware that we will also
work without calculators on a regular basis, and possibly on some exams.

We will use a java-based slope field viewer which I will direct you to on the web page.
We may also occasionally make use of Mathematica, a powerful computer program for doing
mathematics. Mathematica is available on the F&M software server, so feel free to take a
look at it and explore its capabilities. Many of you will have seen it in calculus 3; if and
when we do use it, I will provide instructions.

Finally, course materials will be posted on the web, and we will use email and eDisk for
communication outside of class. Please check your email regularly.

Attendance: I expect you to attend class every day, and you will be held accountable for
everything said and done in class. When you are in class, you should be there mentally as
well as physically. This means you should be alert, have done the reading, and be prepared
to ask questions and work effectively.

I do not intend to take attendance explicitly, but if you are repeatedly absent, I will contact
you to see whether there is a problem I can help you solve. If excessive absences persist, I
may assess a grade penalty.

Missing exams: Do not miss exams. Rescheduling exams is difficult and disruptive, and
will only be considered with persuasive documentation from Dean O’Day or Appel Health
Services.

Contacting me and getting help

Every mathematician finds work at some level difficult – it is normal to get stuck sometimes.
I hope you feel comfortable asking me for help.

If you are standing right in front of me, it is easy to contact me. If you are not, then you
can either find me (thereby reducing the problem to one you know how to solve), send me
email, or call me.

Finding me: I will be in my office and happy to talk to you during my office hours, and
often at other times as well – including evenings, sometimes. If my door is open, please come
in.

Email: I use it, and it is probably the most reliable way to get in touch with me.

Phone: I use it, too. Have been for years. My office phone number is 358-7163, or just
extension 7163 on campus. My home phone number is 391-2944, but please save that for
emergencies.

Other help: The Mathematics Department also has student tutors available. They generally
work 3:00 – 6:00 and 7:00 – 10:00, Sundays through Thursdays, beginning in the second
week of classes. I will notify you of the precise rooms and schedule when they are determined.
Please note: only some of the tutors have enough expertise to help with advanced classes.
The schedule should indicate which is which.

Last words

Curiosity above and beyond the call of duty is strongly encouraged. Honesty is required.
Help is available. Good luck.