Math 2243: Linear Algebra and Differential Equations 

Lecture 020, Fall 2004

Science Classroom Building 175, 11:15 am-12:05 pm MWF

Contact Information for the Instructor:  

Instructor: Willard Miller
Office: Vincent Hall 513
Office Hours: 10:10-11:00 MW, 12:20-1:10 F,  or  by appointment  
Phone: 612-624-7379,

 Discussion Sections:

                 -021  10:10am-11:00am TTh,   VinH 311       Matthew Dobson  Phone:  5-5099,   Office:  VinH 420,

                 -022  10:10am-11:00am TTh,   VinH 211       Ji Hoon Ryoo,  Phone:  5-8553, Office: VinH 422,

                 -023  11:15am-12:05pm TTh,   FolH 334       Matthew Dobson  Phone:  5-5099,   Office:  VinH 420,

                 -024  11:15am-12:05pm TTh,   VinH 311        Ji Hoon Ryoo  Phone:  5-8553, Office: VinH 422,

Brief Course Description

Credit will not be granted if credit has been received for: MATH 2373, prereq 1272 or 1282 or 1372 or 1572,  4 credits

Overview: The course is divided into two somewhat related parts.
Linear algebra: matrices and matrix operations, Gaussian elimination, matrix inverses, determinants, vector spaces and subspaces, dependence, Wronskian, dimension, eigenvalues, eigenvectors, diagonalization.
ODE: Separable and first-order linear equations with applications, 2nd order linear equations with constant coefficients, method of undetermined coefficients, simple harmonic motion, 2x2 and 3x3 systems of linear ODE's with constant coefficients, solution by eigenvalue/eigenvectors, nonhomogenous linear systems; phase plane analysis of 2x2 nonlinear systems near equilibria.

Audience: Part of the standard 2nd year calculus course for students outside of IT.

Text: Farlow, Hall, McDill, West. Differential Equations and Linear Algebra   We will cover Chapters 1-7 (up to Section 7.2) as well as Sections 8.1, 8.2 and 8.5.

4 credits. 3 lectures, 2 recitations per week.
Information about Course Management

Final Grade: Based on a possible 700 points
        Homework and quizzes in recitation sessions -- 100 points
        Three midterms (100 points each: Sept. 28, Oct. 26, Dec. 2) -- 300 points
        Final Exam (Dec. 16)-- 300 points

Effort required: About 12 hours a week
Required Homework: Assigned in recitation session; a subset of the suggested homework.

Calculator and Exam policy: A calculator is useful for homework. Calculators will be allowed but no books or notes in the midterms and final exams.
Makeup policy:  No makeup tests without rigorous emergency reasons. Athletes please present "Proofs of Activities" in advance.
Drop Dates: Students may drop the course without permission before the end of the 8th week of the semester (November 1). If you drop before the end of the 2nd week, no mention of the course appears on your transcript. Otherwise, you receive a "W." Starting November 2,  you need permission to drop the course.

Student Conduct:
  Statement on Scholastic Conduct: Each student should read the college bulletin for the definitions and possible penalties for scholastic dishonesty.  Students suspected of cheating will be reported to the Scholastic Conduct Committee.

FINAL EXAM:  1:30 - 4:30 pm, Thursday, December 16

"All students must have their official University I.D. Card with them at the time of the final exam and must show it to one of the proctors when handing in their exam. The proctor will NOT accept a final exam from a student without an I.D. Card." 

   Syllabus and Suggested Homework

     Date                                Lecture                                    Suggested Homework Problems

Wednesday, September 8
Section 1.1 Dynamical Systems, Modeling
Friday, September 10
Section 1.2 Solutions and Direction Fields
Monday, September 13
Section 1.3 Separation of Variables: Quantitative Analysis
Wednesday, September 15
Section 1.4 Euler's Method: Numerical Analysis
Friday, September 17
Section 1.5 Picard's Theorem: Theoretical Analysis
Monday, September 20
Section 2.1 Linear Equations: The Nature of Their Solutions
Wednesday, September 22
Section 2.2 Solving the 1st-Order Linear ODE
Friday, September 24
Section 2.3 Growth and Decay Phenomena
Monday, September 27

Tuesday, September 28
Midterm I

Wednesday, September 29
Section 2.4 Linear Models: Mixing and Cooling
Friday, October 1
Section 2.5 Nonlinear Models: Logistic Equation
Monday, October 4
Section 2.6 Systems of Differential Equations: A First Look
Wednesday, October 6
Section 3.1 Matrices: Sums and Products
Friday, October 8
Section 3.2 Systems of Linear Equations
Monday, October 11
Section 3.3 The Inverse of a matrix
Wednesday, October 13
Section 3.4 Determinants and Cramer's Rule
Friday, October 15
Section 3.5 Vector Spaces and Subspaces
Monday, October 18
Section 3.6 Basis and Dimension

Wednesday, October 20
Section 3.6
Friday, October 22
Section 4.1 The Harmonic Oscillator
Monday, October 25

Tuesday, October 26
Midterm II, Covers material through Section 3.6

Wednesday, October 27
Section 4.2 Real Characteristic Roots
Friday, October 29
Section 4.3 Complex Characteristic Roots
Monday, November 1
Section 4.4 Undetermined Coefficients
Wednesday, November 3
Section 4.5 Forced Oscillations
Friday, November 5
Section 4.6 Conservation and Conversion
Monday, November 8
Section 5.1 Linear Transformations
Wednesday, November 10
Section 5.2 Properties of Linear Transformations
Friday, November 12
Section 5.3 Eigenvalues and Eigenfunctions
Monday, November 15
Section 5.4 Coordinates and Diagonalization
Wednesday, November 17
Section 6.1 Theory of Linear DE Systems
Friday, November 19
Section 6.2 Linear Systems with Real Eigenvalues
Monday, November 22
Section 6.3 Linear Systems with Nonreal Eigenvalues
Wednesday, November 24
Section 6.4 Decoupling a Linear DE System
Monday, November 29
Section 6.5 Stability and Linear Classification
Wednesday, December 1

Thursday, December 2
Midterm III, covers material through Section 6.4

Friday, December 3
Section 7.1 Nonlinear Systems
Monday, December 6
Section 7.2 Linearization
Wednesday, December 8
Section 8.1 Linear Nonhomogeneous problems
Friday, December 10
Section 8.2 Variation of Parameters
Monday, December 13
Section 8.5  Chaos in Forced Nonlinear Systems
didn't cover this
Wednesday, December 15

Thursday, December 16
Final Exam 1:30-4:30 pm        Sec. 021: VinH 113,
 Secs. 022-023:   MurphyH 130,          Sec. 024:  MurphyH 214

PDF files of exams and practice exams from earlier versions of this course.
Note: These exams were written by different instructors and were sometimes based on different texts, so they should be used with care.

     Midterm1    Midterm 2    Midterm 3    Sample Final    Final

     Exam 1           Exam 2        Exam 3

    Midterm 1a    Midterm 2a    Midterm 3a    Midterm 4a   Final a

   Midterm 1b     Midterm 2b    Midterm 3b    Final b    Final b solns

   Final 1c solns

    My own practice exams and actual exams, based on the present course text.

    Practice Midterm1   Midterm1 solutions     Practice Midterm2    Midterm2  solutions       Practice Midterm3     Midterm3  solutions

     An example of a linear operator on a space of polynomials         Practice Final

Advisory grades for Midterm 1                

85-100  A                                                           Mean      59.46
80-84    A-                                                         Median   55
75-79    B+                                                         STDev    18.90
70-74    B                                                           Number 137
65-69    B-   
55-64    C+  
45-54    C  
40-44    C-  
35-39    D+ 
30-34    D   
< 30       N                                                                                                                                                                               

Advisory grades for Midterm 2  

95-100   A                                                        Mean       76.16
90-94     A-                                                      Median    80
85-89     B+                                                      STDev     20.7
80-84     B                                                        Number  134
75-79     B-
70-74     C+
65-69     C
60-64     C
55-59     C
50-54     C-
45-49     D+
40-44     D
35-39     D
<    34     N
Advisory grades for Midterm 3

80-100    A                                                    Mean       51
70-74      A-                                                  Median    48
65-69      B+                                                  STDev     17.8
55-64      B                                                    Number 127
50-54      B-
45-49      C+
35-44      C
30-34      C-
20-29      D
<   14       N

Miscellaneous Goodies

Plots of direction fields using MAPLE, showing some of the advantages and pitfalls of graphics software packages

            This page also shows a plot of the phase diagram for a linear system of differential equations, considered in class, and a 2D preditor-prey                     model, showing cyclical behavior around an equilibrium point. This particular preditor-prey model system doesn't have an explicit solution    in terms of standard functions. There is also a phase plane diagram for an underdamped harmonic oscillator showing the "black hole" at the     equilibrium point.

Spreadsheet with examples of the Euler method for approximating solutions of 1st order ODEs, including effect of step size

        There are two examples treated:
1. Solution of equation y'=t-3y from t=0 to t=1, with initial condition y(0)=1, step size h=.001     
2. Solution of equation y'=y/t -1  from t=1 to t=2, with initial condition y(1)=1, step sizes h=0.1, h=0.01, h=0.001. According            to theory,  decreasing step size by a  factor of 10  should decrease maximum discretization error by the same factor. However,  decreasing  step size by a factor of 10 may increase maximum roundoff error 10 times. In this example the total error, discretization  and roundoff, is listed in the righthand column. For h=0.1  the maximum error is about 5 x 10-2 , for h=0.01 the maximum error is about 5 x 10-5 , while for for h=0.001 the maximum total error is  5 x 10-4 . Thus reducing step size improves  accuracy initially, but eventually the increased roundoff error actually reduces the accuracy.                                    
Spreadsheet showing Future Value of $100 at 5% annual interest with various periods of compounding, including continuous compounding

Introduction to MATLAB (courtesy of Professor Peter Olver)      Postscript file           PDF file

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