# Courses

## 01:640:252 - Elementary Differential Equations

### Textbook

Textbook:  For current textbook please refer to our Master Textbook List page

### Syllabus

suggested syllabus is available. Dr. Sontag's section can find that syllabus and course material's in the course's Sakai website!

### Supplements

Available supplements, in a uniform PDF format are collected here.

• N1 Introduction to first order equations, with an emphasis on modeling.
• RTB1  Euler's method, including error estimate and application to existence and uniqueness of solutions.
• N2  Some comments on bifurcations.
• N3  Some Remarks on Phase Planes.
• N4  Introduction to Matrix Exponentials.
• RTB2 An easily remembered formula for exponentials of matrices with complex eigenvalues.
• BW1 The method of variation of parameters for solving inhomogeneous systems.

### Math 252 Syllabus for Third Edition of Blanchard, Devaney and Hall

Students in Section 2, Spring 06, should instead use the syllabus linked from that section's webpage
This syllabus is intended as a general outline of the course. It was originally written by E. Sontag for the first edition of the text and adapted to the second edition by R. Wheeden and to the third edition by E. Sontag (Jan 06). Individual instructors may alter its pace, assign different homework, and add or delete topics. Some variations will be described briefly in the notes following the syllabus. Note: The main difference between the 2nd and 3rd Editions is that old Section 1.8 is now "1.9", and a new section 1.8 has been inserted. In addition, many homework problems have been renumbered, and some new problems have been inserted.

Assignments usually refer to sections of the textbook. A designation such as N4 is a link to a supplementary note.

# SectionsSubjectsAssignmentsNotes
1 N1 Modeling all (  answers )
2 1.1 Modeling (continued) 3, 5, 15, 17, 19, 21.
1.2 Separation of Variables 1, 3, 7, 13
3 1.2 Separation of Vars (continued) 25, 29, 31, 35.
4 1.3 Slope Fields all odd 1-13, 14, 15, 17. a
5 1.4 Euler's Method 1, 13, 15.
1.5 Existence and Uniqueness 1, 3, 5, 7, 10. b
1.6 Equilibria and Phase Line 1, 3, 5, 7, 13, 15, 23, 25, 27, 31, 33, 37, 39, 43.
6 N2 Bifurcations (No exercises in N2)
1.7 1, 3, 5, 9, 11, 17.
7 1.8 Linear Differential Equations all odd 1-13, 21, 23.
1.9 Integrating Factors all odd 1-11, 21, 23.
8 2.1 Modelling via Systems 1, 2, 7, 8, 9, 17, 19, 21, 23, 25, 26, 27, 29. c
9 2.2 Geometry of Systems all odd 1-27. a
10 2.3 Analytic Methods all odd 1-11, 19. d
2.4 Euler's Method 1, 3, 5, 14, 15.
11 N3 Phase Plane all ( answers ) e
12 exam 1 Through 2.2 included
13 3.1 Linear Systems all odd 1-9, 13, 17, 19, 21, 27, 29, 33, 35. f
N4 Matrix Exponentials   g
14 N4 Matrix Exponentials (continued) all ( answers ) g
15 3.2 Straight-Line Solutions all odd 1-19
16 3.3 Phase Plane: Real Eigenvalues all odd 1-15.
17 3.4 Phase plane: Complex Eigenvalues all odd 1-15, 19, 21, 23.
18 3.5 Repeated and Zero Eigenvalues all odd 1-17.
19 3.7 The Trace-Determinant Plane
(emphasizing one-parameter families)
parts "c" of: 3, 7, 11, 13. h
20 3.6 Second-Order Linear all odd 13-29; 36(a,b). h,i
21 3.8 3-Dim Linear 4, 5, 6, 7.
22 4.1 Forced Harmonic Oscillators all odd 1-41.
4.2 Sinusoidal Forcing odd 1-13, 17, 27.
23 exam 2 2.3/3.7 (lectures 10/20)
24 4.4 Steady State Special exercises. j
25 4.3 Resonance all odd 1-17, 21
26 5.1 Equilibria, Linearization all odd 1-17, except 5. h
27 8.1 Discrete Systems all odd 1-9, 15, 19, 23, 27, 31.
28 8.2 Fixed/Periodic points 1, 7, 9, 13, 15. k

29 final exam all material covered during the semester

Notes:
a. For numerical assignments, there is a package available in the CD ROM that comes with the book. A strongly suggested alternative is the Java Applet, JOde, which runs on any Java-enabled browser, including those at the University computer labs. Assignments using JOde will be posted to the Section 2, Spring 06 webpage

b. The existence and uniqueness theorem may be applied in abutting regions with continuity across the boundary to allow for piecewise continuous forcing functions.  Projects exploring this have been used in the course.

c. Sections 2.1/2.2 are not really different, and should studied (and possibly lectured upon) simultaneously.  Even 2.3 and 2.4 are not very different, actually.

d. The material on damped harmonic oscillator does not fit well with the topic of section 2.3, and may be deferred until the topic is considered in more detail in chapter 4.

e. Instead of the emphasis on exact trajectories in N3 and related supplements, instructors may introduce isoclines at this point to help guess phase plane portaits in simple cases like saddle points. The aim should be to complement the study of straight line solutions to appear in section 3.2 rather than to insert all of section 5.2 into the syllabus at this point.

f. Note to students: please make sure to review eigenvalues and eigenvectors from your linear algebra notes (which you kept from when you took the course!)

g. Some instructors may wish to skip the notes N4. The matrix exponential, while a useful topic (developed further in notes elaborating on the case of complex eigenvalues ) , may be omitted. The time saved could be used to introduce variation of parameters .

h. Instructors may wish to introduce some or all of section 5.1 when discussing sections 3.6 and 3.7. Students should notice that phase planes for linear systems help predict those for nonlinear ones. Section 5.1 is an important part of the course; if it is not introduced in connection with sections 3.6 and 3.7, instructors should be sure to give adequate coverage later.

i. Problem 36(c) is worth looking at - the design of active automobile suspension systems is an area of much current research (at places like Ford, for example) - this question can be taken as an open ended one - be creative, and perhaps introduce nonlinear damping and nonlinear springs!

j. The exercises for 4.4 are to write the steady-state solution of the odd problems 1-9 of section 4.2 in the form   A cos(wt+f).

k. Instructors emphasizing bifurcations should aim to allow more time for chapter 8 in order to include sections 8.3 and possibly also 8.4.