Math 285. Differential Equations
Syllabus for Instructors

Text: Edwards and Penney, Differential Equations with Boundary Value Problems: Computing and Modelling, Custom Edition for the University of Illinois at Urbana-Champaign, Pearson Custom Publishing, 2008.

Note: This custom edition is taken from Edwards and Penney, Differential Equations with Boundary Value Problems: Computing and Modelling, Fourth Edition. It is identical to the fourth edition except that Chapters 5, 6, 7, and 8 have been removed. If students would prefer to use the full fourth edition, this should not be a problem. The full fourth edition is the standard text for Math 286.

Chapter 1. First Order Differential Equations (6 lectures)

1.1 Differential Equations and Mathematical Models

1.2 Integrals as General and Particular Solutions

1.3 Slope Fields and Solution Curves (3) (Emphasize the existence/uniqueness theorem, and the geometric interpretation and applications of slope fields.)

1.4 Separable Equations and Applications (The material on natural growth and decay is covered in Math 220/221 and can be skipped or quickly reviewed.)

1.5 Linear First Order Equations

1.6 Substitution Methods and Exact Equations (3) (The material on exact equations should be de-emphasized.)

Chapter 2. Mathematical Models and Numerical Methods (2 lectures)

2.1 Population Models

2.3 Acceleration-Velocity Models (Cover one of these in detail. The other can be covered briefly, time permitting.)

Chapter 3. Linear Equations of Higher Order (14 lectures)

3.1 Introduction: Second-Order Linear Equations

3.2 General Solutions of Linear Equations (3) (Emphasize the second order case but introduce the idea of linear independence and the Wronskian for higher order equations.)

3.3 Homogeneous Equations with Constant Coefficients (2) (Include factorization of constant coefficient operators.)

3.4 Mechanical Vibrations (2)

3.5 Nonhomogeneous Equations and Undetermined Coefficients (3) (Include variation of parameters.)

3.6 Forced Oscillations and Resonance (2)

3.8 Endpoint Problems and Eigenvalues (2) (May be covered between 9.4 and 9.5 instead)

Chapter 9. Fourier Series Methods (12 lectures)

9.1 Periodic Functions and Trigonometric Series (3)
(Strongly emphasize the concept of orthogonality. This will be good preparation for students subsequently taking Math 442.)

9.2 General Fourier Series and Convergence (1)

9.3 Fourier Sine and Cose Series (1)

9.4 Applications of Fourier Series (1)

3.8 Endpoint Problems and Eigenvalues (2) (Section 3.8 should be covered now if it was not covered in Chapter 3.)

9.5 Heat Conduction and Separation of Variables (2)

9.6 Vibrating Strings and the One-Dimensional Wave Equation (2)

9.7 Steady-State Temperature and Laplace's Equation (3) (Covers the Dirichlet problem for the disk. Provides another example of substitution methods, this time for a PDE.)

Chapter 10. Eigenvalues and Boundary Value Problems (5 lectures)

10.1 Sturm-Liouville Problems and Eigenfunction Expansions (2)

10.2 Applications of Eigenfunction Series (2)

10.3 Steady Periodic Solutions and Natural Frequencies (1)

Examinations, review and leeway (5 lectures)

Total: 44 lectures