Contact Information for the Instructor:
Instructor: Willard Miller
Office: Vincent Hall 513
Office Hours: 10:1011:00 MW, 12:201:10
F, or by appointment
Phone: 6126247379
miller@ima.umn.edu, miller@math.umn.edu
www.ima.umn.edu/~miller/
Discussion Sections:
021 10:10am11:00am TTh, VinH 311 Matthew Dobson Phone: 55099, Office: VinH 420, dobson@math.umn.edu
022 10:10am11:00am TTh, VinH 211 Ji Hoon Ryoo, Phone: 58553, Office: VinH 422, jhryoo@math.umn.edu
023 11:15am12:05pm TTh, FolH 334
Matthew Dobson
Phone: 55099, Office: VinH 420,
dobson@math.umn.edu
024
11:15am12:05pm TTh, VinH 311
Ji Hoon
Ryoo Phone: 58553, Office: VinH 422,
jhryoo@math.umn.edu
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 firstorder 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 17 (up to Section 7.2) as well as Sections 8.1, 8.2 and 8.5. 4 credits. 3 lectures, 2 recitations per week. 
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 
#2327 
Friday, September
10 
Section 1.2 Solutions
and Direction Fields 
#2,3,6,7,1318,20,23,24,26,29,30 
Monday, September
13 
Section 1.3 Separation
of Variables: Quantitative Analysis 
#12,13,1626,29,31,32,39,40,4347,53,54 
Wednesday, September
15 
Section 1.4 Euler's
Method: Numerical Analysis 
#1,710,12 
Friday, September
17 
Section 1.5 Picard's
Theorem: Theoretical Analysis 
#24,712,21,23 
Monday, September
20 
Section 2.1 Linear
Equations: The Nature of Their Solutions 
#712 
Wednesday, September
22 
Section 2.2 Solving
the 1stOrder Linear ODE 
#16,915,19,20,25,26,3035,45,46 
Friday, September
24 
Section 2.3 Growth
and Decay Phenomena 
#15,7,14,16,17,21,24,26,33 
Monday, September
27 
Review 

Tuesday, September
28 
Midterm I 

Wednesday, September
29 
Section 2.4 Linear
Models: Mixing and Cooling 
#2,4,7,8,1113,1719 
Friday, October
1 
Section 2.5 Nonlinear
Models: Logistic Equation 
#1318,2124 
Monday, October
4 
Section 2.6 Systems
of Differential Equations: A First Look 
#13,57,11,1417,23 
Wednesday, October
6 
Section 3.1 Matrices:
Sums and Products 
#4,5,3134,40,42,4446,51,54,61 
Friday, October
8 
Section 3.2 Systems
of Linear Equations 
#510,2933,35 
Monday, October
11 
Section 3.3 The
Inverse of a matrix 
#710,14,17,18,24,25 
Wednesday, October
13 
Section 3.4 Determinants
and Cramer's Rule 
#1,47,11,12,1618,20,23,2932,3436 
Friday, October
15 
Section 3.5 Vector
Spaces and Subspaces 
#2,46,8,9,1214,1620,22,23,25,26,34,3638,4351,56 
Monday, October
18 
Section 3.6 Basis
and Dimension 

Wednesday, October
20 
Section 3.6 
#1,2,58,1114,1618,2022,2632,38,39,4245,4750,5458,64,65 
Friday, October
22 
Section 4.1 The
Harmonic Oscillator 
#46,810,1416,21,22,2528,3139,49,52 
Monday, October
25 
Review 

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

Wednesday, October
27 
Section 4.2 Real
Characteristic Roots 
#1624,2830,35,3841,43,47,55,56,5962 
Friday, October
29 
Section 4.3 Complex
Characteristic Roots 
#13,1528,3436,41,42 
Monday, November
1 
Section 4.4 Undetermined
Coefficients 
#46,2233,36,39,40 
Wednesday, November
3 
Section 4.5 Forced
Oscillations 
#413,1517,20,22,23 
Friday, November
5 
Section 4.6 Conservation
and Conversion 
#1,2,813,1720,32,33,40,41,44 
Monday, November
8 
Section 5.1 Linear
Transformations 
#610,1315,1729,33,34,3743,47,59,6163,6567,7073,7880 
Wednesday, November
10 
Section 5.2 Properties
of Linear Transformations 
#2,3,6,7,1014,23,30,34,35,38,39,4447,5052,5561 
Friday, November
12 
Section 5.3 Eigenvalues
and Eigenfunctions 
#69,15,18,20,22,24,2629,31,3336,3941,4547 
Monday, November
15 
Section 5.4 Coordinates
and Diagonalization 
#13,79,1315,2023,2733,3941,4344 
Wednesday, November
17 
Section 6.1 Theory
of Linear DE Systems 
#3,5,7,9,1719,2326,3335 
Friday, November
19 
Section 6.2 Linear
Systems with Real Eigenvalues 
#1523,28,31,39,40,43 
Monday, November
22 
Section 6.3 Linear
Systems with Nonreal Eigenvalues 
#1417,2327,30,33 
Wednesday, November
24 
Section 6.4 Decoupling
a Linear DE System 
#35,911,15,16 
Monday, November
29 
Section 6.5 Stability
and Linear Classification 
#112,14,15,17 
Wednesday, December
1 
Review 

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

Friday, December
3 
Section 7.1 Nonlinear
Systems 
#11,13,15,17,19,21,23,2733,35,37 
Monday, December
6 
Section 7.2 Linearization 
#19,1118,2023 
Wednesday, December
8 
Section 8.1 Linear
Nonhomogeneous problems 
#3,5,11,13,1517,24,25 
Friday, December
10 
Section 8.2 Variation
of Parameters 
#39,1215,18,19 
Monday, December
13 
Section 8.5 Chaos
in Forced Nonlinear Systems 
didn't cover this 
Wednesday, December
15 
Review 

Thursday, December
16 
Final Exam 1:304:30
pm Sec. 021: VinH 113, 
Secs. 022023:
MurphyH 130, Sec. 024: MurphyH
214 
1. Solution of equation y'=t3y 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.