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\centerline{\bf MATHEMATICS 2250 SAMPLE SYLLABUS}
\centerline{\bf Ordinary Differential Equations and  Linear Algebra}
\centerline{Spring semester 2001}
\bigskip
\line{ \vbox{
  \hbox{\bf texts:\qquad}
  \hbox{}
  \hbox{}
  \hbox{\bf when:}
  \hbox{\bf where:}}
 \vbox{
  \hbox{\it Differential Equations \& Linear Algebra,}
  \hbox{and {\it Computer Projects} supplement}
  \hbox{\qquad by C.H. Edwards Jr. and David E. Penney}
  \hbox{MWF 11:50-12:40 }
  \hbox{JFB 102}}
 \hfill}
\medskip
\line{ \vbox{
  \hbox{\bf instructor:}
  \hbox{\bf \qquad office:}
  \hbox{\bf \qquad telephone:}
  \hbox{\bf \qquad email:}
  \hbox{\bf \qquad office hours:\qquad}}
 \vbox{
  \hbox{Prof. Nick Korevaar}
  \hbox{JWB 218}
  \hbox{581-7318}
  \hbox{korevaar{\@}math.utah.edu}
  \hbox{M 2-2:50 p.m.,\ T 12:30-2:30 p.m.,\  
    W 2-2:50 p.m.,\ F 8:20-9:00 a.m.}}
 \hfill}
\medskip\noindent
\line{{\bf course home page:}\qquad www.math.utah.edu/
$\sim$korevaar/2250spring01.html\hfill}
\bigskip\noindent
{\bf prerequisites:}  Math 1210 and 1220 or the equivalent (first-year
Calculus, with a very brief introduction to 
linear differential equations).   In addition you
are expected to be familiar with vectors, curves, veclocity (tangent),
and acceleration vectors, either from Physics 2210 or Math 2210, or their
equivalents.
\bigskip
\noindent
{\bf course outline:}
This course is an introduction to differential equations, and how they
are used to model problems arising in engineering and science.
 Linear algebra is introduced as a tool for analyzing systems of
differential equations, as well as standard linear equations.
 Computer
projects will be assigned to enhance the material.

We will cover most of chapters 1-7 and 10 in the Edwards-Penney text.  
The course begins with first order differential equations, a subject
which you touched on in Calculus. Recall that a differential
equation is an equation involving an unknown function and its
derivatives, that such equations often arise in science, and that
the order of a differential equation is defined to be the highest order
derivative occuring in the equation.
The goal is to understand the solution functions to differential
 equations since these functions will be describing
the scientific phenomena which led to the
differential equation in the first place.
  In chapters one and two of Edwards-Penney
 we learn analytic techniques for
solving certain first order DE's, the geometric meaning of graphs
of solutions, and the numerical techniques for approximating solutions
which are motivated by this geometric interpretation.  We will carefully
study the logistic population growth model from mathematical biology and
various velocity-acceleration models from physics.

At this point in the course we will take a four week digression 
from differential equations, to learn the
fundamentals of linear algebra in chapters 3-4.
You need a basic understanding of this  field of mathematics in 
order to talk meaningfully about the
theory of higher order linear DE's and of
systems of linear DE's.  Chapter 3 starts out with matrix equations and
the Gauss-Jordan method of solution.  When you see such equations in
high-school algebra you might be thinking of intersecting lines in the
plane, or intersecting planes in space, or ways to balance chemical reactions,
but the need to understand generally how to solve such equations is
pervasive in science.  From this concrete beginning we study abstract
vector spaces in chapter 4.  This abstract theory is useful not only
to the understanding of Euclidean space $\bold R^n$,
but it is also the framework which allows us to understand solution spaces
to systems of linear
differential equations.  

In chapter 5 we will study the theory
of second order and higher-order linear DE's,  and focus on  the 
 ones which describe basic mechanical
 vibrations.  You were introduced to these equations
in Calculus, and we will treat them more
completely now, including a careful study of forced oscillations
and resonance.  

To succesfully model more complicated physical systems like
shaking in multi-story buildings, one needs
the framework of linear systems of differential equations.  
The linear algebra of eigenvalues and eigenvectors is
developed in chapter 6 as a tool for studying systems
of DE's in chapter 7, where for example eigenvalues and
eigenvectors will be related to fundamental modes for
mass-spring systems.

Our final course topic is the Laplace transform and its applications to
the study of linear DE's, in chapter 10.
  This ``magic'' transform takes differential
equations and ``transforms'' them into algebraic equations.  You will
have to see it in action to appreciate it. You will see, for example, 
that this method
gives a powerful way to study forced oscillations in the physically
important cases that the forcing terms
are square waves or other discontinuous functions.
Electrical engineers often use Laplace transform techniques to
 understand circuits.

\smallskip
\noindent
{\bf computer projects:}
There will be three computer projects assigned during the
semester, related to
the classroom material.  They will
be written  in the software
package MAPLE. In addition, you will be asked to use this
computer software to check various homework calculations
from the
linear algebra and Laplace transform sections of the course.
There is a Math Deparment Computer
Lab at which you all automatically have accounts, and
there are other labs around campus where Maple is also available,
for example at the College of Engineering.  There will
tutoring center support for these projects (and for your other
homework) as well.  There will be introductory sessions
to the lab and to Maple, and these will be announced in class.
The Math Department Lab is currently located in South Physics
205.  More information about 
the lab can be found at the Math
Department web page for undergraduates, www.math.utah.edu/ugrad .

\smallskip\noindent
{\bf tutoring/help sessions:}
There will be special help sessions for Math 2250 students, and
these will be held
on Tuesdays. (Homework will be due on Wednesdays.)
  Currently the sessions are scheduled as follows, although
this may be changed:  In JWB 335 from 7:30-8:20 a.m.,
 in JTB 140 from 8:35-9:25 a.m.; in EMCB 114 from
9:40-10:30 a.m., in JTB 130 from 4:30-5:30 p.m., in JFB 102
from 5:30-7:30 p.m.
 Students may attend as many or as few of these sessions
as they wish.  One or more of the morning sessions may
be reserved for a special Tuesday lecture, e.g. February 6.
In addition to the special 2250 sessions, the Math Department
Tutoring Center is located in Mine 210,
and is open for free tutoring from 8 a.m. to 8 p.m. on M-Th, and from 
8 a.m. to 2 p.m. on Friday.  Some, but not all of the math tutors
welcome questions from Math 2250 students.  To see the times and
specialities of various tutors, consult the web address
www.math.utah.edu/ugrad/tutoring.html .


\smallskip
\noindent
{\bf grading:}\qquad There will be two midterms, a comprehensive final 
examination, and homework.  Each midterm will count for 20\% of your grade,  
homework (from the book and
on the computer) will count for 30\%, and the final exam will make up the 
remaining 30\%.  The book homework will be assigned daily and collected 
weekly, on Wednesdays. 
The three Maple project assignment and collection dates
are shown on the syllabus.
A homework grader will partially grade your book homework and
Maple assignments. 
The value of carefully working homework problems is that mathematics  
(like anything) must be practiced and experienced to be learned.
\smallskip
It is the Math Department policy, and mine as well, to grant any
withdrawl request until the University deadline of Friday March 2.
\medskip\noindent
{\bf ADA statement:}  The American with Disabilities Act requires 
that reasonable accomodations be provided for students with physical, sensory, 
cognitive, systemic, learning, and psychiatric disabilities.  
Please contact me at the beginning of the semester to discuss 
any such accommodations for the course.


\vfill\break
\centerline{\bf Tentative Daily Schedule}
\medskip
\centerline{exam dates fixed, Maple start dates approximate}
\centerline{ daily subject matter approximated}
\bigskip\bigskip
\line{ \vbox{
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{T}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{}\hbox{F}\hbox{}
  \hbox{M}\hbox{T}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{}\hbox{W}\hbox{F}
 }


 \qquad
 \vbox{
  \hbox{8 Jan}\hbox{10 Jan}\hbox{12 Jan}\hbox{}
  \hbox{15 Jan}\hbox{16 Jan}\hbox{17 Jan}\hbox{19 Jan}\hbox{}
  \hbox{22 Jan}\hbox{24 Jan}\hbox{26 Jan}\hbox{}
  \hbox{29 Jan}\hbox{31 Jan}\hbox{}\hbox{2 Feb}\hbox{}
  \hbox{5 Feb}\hbox{6 Feb}\hbox{7 Feb}\hbox{9 Feb}\hbox{}
  \hbox{12 Feb}\hbox{14 Feb}\hbox{16 Feb}\hbox{}
  \hbox{19 Feb}\hbox{21 Feb}\hbox{23 Feb}\hbox{}
  \hbox{26 Feb}\hbox{28 Feb}\hbox{2 Mar}\hbox{}
  \hbox{5 Mar}\hbox{7 Mar}\hbox{9 Mar}\hbox{}
  \hbox{19 Mar}\hbox{}\hbox{21 Mar}\hbox{23 Mar}
  }

 \qquad
 \vbox{
  \hbox{1.1}\hbox{1.2}\hbox{1.3}\hbox{}
  \hbox{none}\hbox{Maple I}\hbox{1.4}\hbox{1.5}\hbox{}
  \hbox{2.1}\hbox{2.2}\hbox{2.3}\hbox{}
  \hbox{2.4 \& 2.6}\hbox{3.1}\hbox{Maple I due}\hbox{3.2}\hbox{}
  \hbox{3.3}\hbox{3.4}\hbox{3.5}\hbox{3.6}\hbox{}
  \hbox{3.6}\hbox{exam 1}\hbox{4.1}\hbox{}
  \hbox{none}\hbox{4.2}\hbox{4.3}\hbox{}
  \hbox{4.4}\hbox{4.5}\hbox{5.1}\hbox{}
  \hbox{5.2}\hbox{5.3}\hbox{5.4}\hbox{}
  \hbox{5.5}\hbox{Maple II}\hbox{5.5-5.6}\hbox{5.6}
  }

 \qquad
 \vbox{
  \hbox{modeling and differential equations}
  \hbox{integral solutions}
  \hbox{direction fields and solution curves}
  \hbox{}
  \hbox{Martin Luther King Day}
  \hbox{{\it begin} Maple tutorial and first project}
  \hbox{separable equations}
  \hbox{first order linear DE's}
  \hbox{}
  \hbox{population models}
  \hbox{equilibrium solutions and stability}
  \hbox{acceleration-velocity models}
  \hbox{}
  \hbox{numerical techniques: Euler and Runge Kutta}
  \hbox{introduction to linear systems}
  \hbox{{\it hand in} first Maple project}
  \hbox{matrices and Gaussian elimination}
  \hbox{}
  \hbox{reduced row-echelon form}
  \hbox{matrix operations}
  \hbox{inverses of matrices}
  \hbox{determinants}
  \hbox{}
  \hbox{determinants}
  \hbox{1-3}
  \hbox{the vector space $\bold R^3$}
  \hbox{}
  \hbox{Presidents' Day}
  \hbox{the vector space $\bold R^n$ and subspaces}
  \hbox{linear combinations and linear independence}
  \hbox{}
  \hbox{bases and dimension of vector spaces}
  \hbox{general vector spaces}
  \hbox{second-order linear differential equations}
  \hbox{}
  \hbox{general solutions of linear equations}
  \hbox{homogeneous equations with constant coefficients}
  \hbox{mechanical vibrations}
  \hbox{}
  \hbox{undetermined coefficients and variation of parameters}
  \hbox{{\it begin}  Tacoma narrows}
  \hbox{and unforced oscillations and resonance}
  \hbox{continued}
   }}

 \vfill\pagebreak
\line{
 \vbox{
  \hbox{M}\hbox{W}\hbox{F}\hbox{}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}
  \hbox{M}\hbox{W}\hbox{F}\hbox{}\hbox{}
  \hbox{M}\hbox{W}\hbox{}
  \hbox{T}   }
 \qquad
 \vbox{
  \hbox{26 Mar}\hbox{28 Mar}\hbox{30 Mar} \hbox{}\hbox{}
  \hbox{2 Apr}\hbox{4 Apr}\hbox{6 Apr}\hbox{}\hbox{}
  \hbox{9 Apr}\hbox{11 Apr}\hbox{13 Apr}\hbox{}
  \hbox{16 Apr}\hbox{18 Apr}\hbox{20 Apr}\hbox{}\hbox{}
  \hbox{23 Apr}\hbox{25 Apr}\hbox{}
  \hbox{1 May}   }
 \qquad
 \vbox{
  \hbox{6.1}\hbox{6.2}\hbox{exam 2}\hbox{Maple II due}\hbox{}
  \hbox{7.1}\hbox{7.2}\hbox{7.3}\hbox{Maple III}\hbox{}
  \hbox{7.3-7.4}\hbox{7.4}\hbox{10.1}\hbox{}
  \hbox{10.2}\hbox{10.3}\hbox{10.4}\hbox{Maple III due}\hbox{}
  \hbox{10.4-10.5}\hbox{10.5}\hbox{}
  \hbox{FINAL EXAM}   }
 \qquad
 \vbox{
  \hbox{introduction to eigenvalues}
  \hbox{diagonalization of matrices}
  \hbox{4-6}
  \hbox{{\it hand in} second Maple project}
  \hbox{}
  \hbox{first order systems and applications}
  \hbox{matrices and linear systems}
  \hbox{eigenvalue method for linear systems}
  \hbox{{\it begin} shaking building}
  \hbox{}
  \hbox{\& second order systems and mechanical applications}
  \hbox{continued}
  \hbox{Laplace transforms and inverse transforms}
  \hbox{}
  \hbox{transformation of initial value problems}
  \hbox{translation and partial fractions}
  \hbox{derivatives, integrals, products of transforms}
  \hbox{{\it hand in} third Maple project}
  \hbox{}
  \hbox{\& periodic and piecewise continuous forcing functions}
  \hbox{continued}
  \hbox{}
  \hbox{entire course\  11:30-1:30}   }    }




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