Math 356/756, Elementary Differential Equations


Time and place: TR, 1:25pm - 2:40pm, French Science 2237.

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

Office hours: T 3:00pm - 5:00pm @ 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 Thursdays. 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 February 28. The date of the final is May 3.

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

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

Lecture Date Book Sections Topics
1 1/10 1.1-3, 2.1 Derivatives, integrals, first-order ODE, normal forms, IVP, direction field.
2 1/15 2.2, 2.4 Separable equations, linear equations, integrating factor, variation of parameters.
3 1/17 2.3, 2.5 Models of motion, linear/quadratic air resistance, terminal velocity, scaling, mixing problems
4 1/22 2.6 Differential forms, integral curves, exact differential equations.
5 1/24 2.6, 2.9 Integrating factors, homogeneous equations, autonomous equations, equilibrium points and solutions.
6 1/29 2.9, 2.7 Phase line, stability, existence/uniqueness of solutions, application of uniqueness theorem.
7 1/31 2.8, 4.1 Dependence of solutions on the initial conditions, second order (linear) ODE, structure of the general solutions, Wronskian.
8 2/5 4.3, 4.4 Linear, homogeneous equations with constant coefficients, harmonic motion, undamped harmonic motion, amplitude and phase, damped harmonic motion.
9 2/7 4.5, 4.6 Methods of undetermined coefficients, variation of parameters.
10 2/12 4.7, 8.1, 8.2 Forced harmonic motion, ODE systems, vector notations, geometric interpretation of solutions of an ODE system, phase space plots, direction field.
11 2/14 8.3 - 8.5 Existence and uniqueness, linear systems, linear independence and dependence, Wronskian,
12 2/19 9.1, 9.2 Linear systems with constant coefficients, eigenvalues/eigenvectors, planar systems, Four different cases for solving planar systems.
13 2/21 9.2, 9.3 Four different cases for solving planar systems, phase plane portraits.
14 2/26 N/A Review
15 2/28 N/A Midterm
16 3/5 9.4, 9.5 The trace-determinant plane, higher-dimensional systems, algebraic/geometric multiplicity.
17 3/7 9.6 Exponential of a matrix, trunction, generalized eigenvectors and the corresponding solutions, the solution procedures (for high-dimensional homogeneous linear systems with constant coefficients.)
18 3/19 9.7, 9.8 Qualitative analysis of linear systems, higher-order linear equations, structure of the general solution.
19 3/21 9.8, 9.9 Fundamental set of solutions (for higher-order homogeneous differential equation with constant coefficient), inhomogeneous linear systems, variation of parameters, computing the exponential of a matrix.
20 3/26 10.1 - 10.3 Linearization of a nonlinear systems, characterization of equilibrium points, long-term behavior of solutions, invariant sets
21 3/28 10.3, 10.4 Invariant sets, nullclines, global analysis, limit set, limiting cycles
22 4/2 10.4, 10.5 Limiting graphs, Bendixson alternative, conserved quantity.
23 4/4 12.1 Fourier series, even and odd functions.
24 4/9 12.2, 12.3, 13.1 Convergence of Fourier series, Fourier cosine and sine series, derivation of heat equation.
25 4/11 13.1, 13.2 Derivation of heat equation, solving the homogenous heat equation with Dirichlet boundary condition.
26 4/16 13.2, 13.3 Solving the homogeneous heat equation with Neumann boundary condition, derivation of the wave equation.
27 4/18 13.3 D'Alembert's solution
28 4/23


Homework problems: