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Differential Equations
Math 4027-1
Louisiana State University
Spring Semester, 2013
Prof. Stephen Shipman
Place: Room 111 of Lockett Hall
Time: Tuesday and Thursday from 10:30 to 11:50
Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday 9:00-11:00, Thursday 9:00-10:30, or by appointment
For a PDF version of the basic course information on this page,
click here: 4027syl.pdf.
Textbook
A Second Course in Elementary Differential Equations, by Paul Waltman.
Prerequisite
Math 2057 (Multidimensional Calculus) and Math 2085 (Linear Algebra)
Course Content
The course will cover all four chapters of the textbook. The main topics are
- Chapter 1: Systems of Linear Differential Equations.
Every system of differential equations of any order (the order is the number of derivatives) can be written in first order form (involving first derivatives only) by introducing new quantities (like velocity or acceleration) into the system. In first-order form x'=Ax (x(t) is a vector and A is a matrix), all linear systems can be treated in a systematic way. The main mathematics is linear algebra and the matrix exponential. The eigenvalues of the matrix A of the system determine its behavior, including stability and whether it undergoes oscillations. There are lots of interesting examples from physics and engineering.
- Chapter 2: Two-Dimensional Autonomous Systems.
Almost every physical system is practically affected by nonlinearity, either weakly or strongly. Most of the theory of nonlinear systems is qualitative—"exact" solutions cannot be written down (accurate solutions must be computed numerically). Our analysis will address fundamental ideas such as fixed points and periodic cycles and their stability in the case of two-dimensional systems (two quantities evolving in time) that are autonomous (independent of the starting time). Analysis of stability is based on linear approximation around fixed points and draws on Chapter 1. Examples include the circular pendulum and population dynamics. Three-dimensional (and higher) systems admit more complicated behavior, such as chaos and strange attractors, and Chapter 2 will prepare people to study those things later on.
- Chapter 3: Existence Theory.
This is the most theoretical part of the course. One wants to know if a system of equations really exhibits the behavior understood for the physical system being modeled. This includes existence and uniqueness of solutions. Understanding how to prove these things elucidates aspects of differential equations that prevail in many physical situations, namely non-smoothness of system parameters and unpredictability or controllability of its trajectories.
- Chapter 4: Boundary-Value Problems.
We shift our attention from stipulating initial data of a system to stipulating values at the endpoints of an interval. For example, motions of a string are governed by a differential equation, and clamping the endpoints of the string determines which motions are possible. The frequencies of oscillation are, mathematically, eigenvalues of a linear differential operator. The eigenvalues, and therefore the frequencies, or harmonics, of the string depend on the boundary conditions (for example, the endpoints could be free to move transversely instead of being clamped). Differential equations with periodic coefficients (Ch. 1 §10) can also be studied in this context because periodicity amounts to conditions at two "endpoints".
Assignments
Students will be expected to present their solutions in readable, logically coherent arguments, with proper use of mathematical symbols.
Due Date |
Section |
Problems |
| Jan. 22
| Ch.1 §3
| 2, 3, 4
|
| Jan. 22
| Ch.1 §4
| 3, 4, 5, 6, 8, 9, 10
|
| Jan. 29
| Ch.1 §5
| 3, 4, 9, 10
|
| Jan. 29
| Ch.1 §6
| 2, 5, 9 (Also find a basis of real solutions.)
|
| Jan. 29
| Ch.1 §7
| 1, 2, 7, 8 (Use Jordan canonical form or Putzer algorithm.)
|
| Feb. 5
| Ch.1 §8
| 1bc, 2bc, 4, 5
|
| Feb. 5
| Ch.1 §9
| 1, 8, 10, 11, 12
|
| Feb. 19
| Ch. 2 §2
| 1a, 6
|
| Feb. 26
| Ch. 2 §3-4
| The assignment below
|
| Feb. 26
| Ch. 2 §5
| 2
|
| Mar. 5
| Ch. 2 §5
| 6, 7, 11
|
| Mar. 5
| Ch. 2 §6
| 1a, 2, 3, 6
|
| Mar. 7
| Ch. 2 §7
| 6, 8, 10
|
| Mar. 19
| Ch. 3 §2
| 1, 7, 8
|
| Mar. 26
| Ch. 3 §3
| 3, 4, 5, 7
|
| Mar. 26
| Ch. 3 §4
| 1
|
| For practice
| Ch. 3 §4
| 2, 4
|
| April 23
| Ch. 4 §2
| 1, 4, 5, 8
|
| April 30
| Ch. 4 §4
| 1ab, 2ab
|
| April 30
| Ch. 4 §5
| 5
|
Assignment for §3-4
1. For each of the five matrices below (a-e), determine a nonsingular matrix M and a matrix B in normal form (diagonal, Jordan form, or dilation-rotation) such that AM = MB.
2. In each case (a-e), find exp(Bt) and draw a careful sketch of the phase portrait for the
ODE y' = By.
3. In each case (a-e), draw a careful sketch of the phase portrait for the ODE x' = Ax by considering the fundamental matrix solution Φ(t) = M exp(Bt) and morphing your sketch in (2) from y to x coordinates, where x = My.
a.
b.
c.
d.
e.
Exam schedule
- Exam 1: Thursday, February 7 on Chapter 1
- Exam 2: Tuesday, March 12 on Chapter 2
- Exam 3: Tuesday, April 9 on Chapter 3
- Exam 4: Tuesday, April 30 on Chapter 4
- Final exam: Thursday, May 9, from 5:30 to 7:30 PM in Lockett 111
The final exam will be comprehensive.
Evaluation
A graduate-student grader will grade at least part of the assignments. A subset of the assigned problems will be chosen to be graded. The instructor will grade the exams. Evaluation of performance in the course is computed as follows:
- Assignments: 15%
- Four in-class exams: 60% (15% each)
- Final exam: 25%
The final exam score will replace the lowest in-class exam score whenever the latter is the lower of the two scores.
Grading scale: A---at least 90%; B---at least 80%; C---at least 70%; D---at least 60%.
Ethical Conduct
Students may discuss problems with each other and other people and consult other literature; however, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone. No joint work in any capacity may be submitted for evaluation.
Students must abide by the LSU
Code of Student Conduct.
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