Math 441. Differential Equations
Instructor Syllabus
Textbook: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th edition, John Wiley & Sons, Inc., 2012.
Instructor resource (from library): Polking, Boggess and Arnold (PBA), Differential Equations, 2nd edition, 2006
|
Topic |
Section |
Hours |
|---|---|---|
|
Models, basic examples, classification of DEs |
1.1-1.3, applications from 2.3 |
3 |
|
First order linear and separable DEs, and statement of existence and uniqueness theorems |
2.1, 2.2, 2.4 |
4 |
|
Proof of existence and uniqueness (recommend deferring this material to end of semester, and treating systems of DEs) |
2.8 |
3 |
|
Autonomous equations and population dynamics, and miscellaneous topics e.g. reduction of 2nd order |
2.5 |
2 |
|
Euler’s method |
2.7 |
1 |
|
Second order linear DEs |
3.1-3.6 |
6 |
|
Mechanical oscillators (un/damped, un/forced) and resonance |
3.7, 3.8 |
4 |
|
Higher order linear DEs |
Chapter 4 highlights |
1 |
|
Series solutions near an ordinary point, regular singular point |
5.2, 5.4 |
2 |
|
Introduction to systems, existence and uniqueness, equilibrium points, phase portraits (state in n dimensions, work in 2 dimensions) |
e.g. PBA 8.1-8.5 |
3 |
|
Linear homogeneous systems: real and complex eigenvalues for the 2x2 case, saddles, nodes, spirals, trace-determinant diagram, stability |
e.g. PBA 9.1-9.4 |
4 |
|
Matrix exponentials and Putzer’s algorithm (state in n dimensions, work in 2 dimensions) |
e.g. PBA 9.6 |
2 |
|
Linearization of autonomous systems, long-time behavior, polar coordinates and limit cycles |
e.g. PBA 10.1, 10.4 |
2 |
|
Instructor’s choice (e.g. energy conservation/dissipation, damped pendulum phase portrait, Lorenz system and chaos, competing species) |
e.g. PBA 10.2, 10.5, 10.6 |
3 |
|
Exams |
|
3 |
|
Total |
|
43 |
Comments to instructors
Many students arrive in Math 441 without having taken linear algebra, which is not a prerequisite. Students do know how to multiply and invert 2x2 matrices. So, while the results on systems can be stated in all dimensions, the examples, computations and homework should generally be restricted to 2-dimensions.
The material on systems in Boyce and DiPrima’s textbook (Chapters 7 and 9) emphasizes higher dimensional systems and is rather formal in approach. Instructors can base their lectures for this part of the course on more engaging source material found elsewhere (such as the book by Polking, Boggess and Arnold). The exercises in Boyce and DiPrima do include many 2-dimensional problems that can be assigned.
Phase lines for autonomous DEs in 1-dim should be drawn vertically, to match the direction of the coordinate axis.
Differential equations connect to a multitude of applications e.g. the SI system for epidemics. Students and instructors alike find these applications to be fascinating and enjoyable. When exploring these applications, instructors and students are encouraged to use a free online numerical solution plotter such as http://www.bluffton.edu/homepages/facstaff/nesterd/java/slopefields.html, which can handle first order DEs as well as first order autonomous systems in 2-dimensions.