**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
**

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

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! )

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.

First (office-visiting) assignment: | 2% |

Homework assignments: | 33% |

First midterm | 20% |

Second midterm | 20% |

Final Exam | 25% |

Total | 100% |

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

will only be considered with persuasive documentation from Dean O’Day or Appel Health

Services.

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.

often at other times as well – including evenings, sometimes. If my door is open, please come

in.

extension 7163 on campus. My home phone number is 391-2944, but please save that for

emergencies.

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.