Math 356/756, Elementary Differential Equations


Time and place: WF, 3:05pm - 4:20pm, Biological Sciences 155.

Instructor: Wei Zhu (zhu at math.duke.edu).

Office hours: M, 11:00am - 1:00pm @ Physics 023.

Textbook: Differential Equations with Boundary Value Problems, by Polking, Boggess, and Arnold, second edition. We will cover much of Chapters 1, 2, 3, 4, 8, 9, 10, 12, 13.

Homework: Homework is assigned every week (see the end of this page for the homework problems), and will be collected in class on Fridays. Further information is given below.
  • No late homework will be accepted, but two lowest scores will be dropped.
  • Write your name and ID number at the top of the first page.
  • Staple your pages.
  • You can collaborate on the homework, but the final version must be written in your own words. Copying the solutions of others is strictly forbidden.
Exams: There will be a midterm and a final exam. The date of the midterm is October 19. The date of the final is December 16.

Grades: Homework 20%, midterm 35%, final 45%.

Syllabus: The following table will be updated as we proceed.

Lecture Date Book Sections Topics
1 8/29 1.1-3, 2.1 Derivatives, integrals, first-order ODE, normal forms, IVP, direction field.
2 8/31 2.2, 2.4 Separable equations, linear equations, integrating factor, variation of parameters.
3 9/5 2.3 Models of motion, linear/quadratic air resistance, terminal velocity, scaling.
4 9/7 2.5, 2.6 Mixing problems, differential forms, integral curves, exact differential equations.
5 9/12 2.6, 2.9 Integrating factors, homogeneous equations, autonomous equations, equilibrium points and solutions.
6 9/14 N/A Cancelled due to the weather
7 9/19 2.9, 2.7 Phase line, stability, existence/uniqueness of solutions, application of uniqueness theorem.
8 9/21 2.8, 4.1 Dependence of solutions on the initial conditions, second order (linear) ODE, structure of the general solutions, Wronskian.
9 9/26 4.3, 4.4 Linear, homogeneous equations with constant coefficients, harmonic motion, undamped harmonic motion, amplitude and phase, damped harmonic motion.
10 9/28 4.5, 4.6 Methods of undetermined coefficients, variation of parameters.
11 10/3 4.7, 8.1 Forced harmonic motion, ODE systems, vector notations.
12 10/5 8.2 - 8.5 Geometric interpretation of solutions of an ODE system, phase space plots, direction field, existence and uniqueness, linear systems
13 10/10 8.5, 9.1, 9.2 Linear independence and dependence, Wronskian, linear systems with constant coefficients, eigenvalues/eigenvectors, planar systems.
14 10/12 9.2 Four different cases for solving planar systems.
15 10/17 N/A Review
16 10/19 N/A Midterm
17 10/24 9.3, 9.4 Phase plane portraits, the trace-determinant plane.
18 10/26 9.5, 9.6 Higher-dimensional systems, algebraic/geometric multiplicity, exponential of a matrix and its relation to a higher-dimensional system.
19 10/31 9.6 Exponential of a matrix (continued), trunction, generalized eigenvectors and the corresponding solutions, the solution procedures (for high-dimensional homogeneous linear systems with constant coefficients.)
20 11/2 9.7, 9.8 Qualitative analysis of linear systems, higher-order linear equations, structure of the general solution, fundamental set of solutions (for higher-order homogeneous differential equation with constant coefficient.)
21 11/7 9.9, 10.1 Inhomogeneous linear systems, variation of parameters, computing the exponential of a matrix, linearization of a nonlinear 1D equation.
22 11/9 10.1, 10.2 Linearization of nonlinear systems, characterization of equilibrium points, long-term behavior of solutions, stable, unstable, asymptotically stable.
23 11/14 10.3, 10.4 Invariant sets, nullclines, global analysis, limit set, limiting cycles, limiting graphs.
24 11/16 10.4 - 10.6 Bendixson alternative, conserved quantity, nonlinear mechanics, conserved systems
25 11/28 12.1, 12.2 Fourier series, even and odd functions, convergence.
26 11/30 12.3, 13.1 Fourier cosine and sine series, derivation of heat equation.
27 12/5
28 12/7 Review


Homework problems: