Proposed Math 284. Intro Differential Systems
Syllabus for Instructors

(4 Credit Hours)

(Prepared by R. Muncaster - November 2005)

Text: Edwards and Penney, Differential Equations & Linear Algebra, 2nd Edition, Prentice-Hall, 2005.

Chapter 1. First Order Differential Equations (6 lectures)

1.1 Differential Equations and Mathematical Models (1)

1.2 Integrals as General and Particular Solutions (1)

1.3 Direction Fields and Solution Curves (1)

(Emphasize the existence/uniqueness theorem, and the geometric interpretation and applications of slope fields.)

1.4 Separable Equations and Applications (1)

(The material on exponential growth and decay is covered in Math 220 and can be skipped or quickly reviewed.)

1.5 Linear First Order Equations (1)

1.6 Substitution Methods and Exact Equations (1)

(The material on exact equations has been de-emphasized.)

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

2.1 Population Models

2.3 Acceleration-Velocity Models

(Cover one of these two sections in detail. The other can be covered briefly, time permitting.)

Chapter 3. Linear Systems and Matrices (12 lectures)

3.1 Introduction to Linear Systems (1)

3.2 Matrices and Gaussian Elimination (2)
3.3 Reduced Row-Echelon Matrices (1)
3.4 Matrix Operations (2)
3.5 Inverses of Matrices (3)
3.6 Determinants (2)
4.1 The Vector Space R³ (1)

Chapter 5. Higher-Order Linear Differential Equations (12 lectures)

5.1 Introduction: Second-Order Linear Equations (1)

5.2 General Solutions of Linear Equations (2)

(Emphasize the second order case but introduce the idea of linear independence and the Wronskian for higher order equations.)

5.3 Homogeneous Equations with Constant Coefficients (2)

5.4 Mechanical Vibrations (2)

5.5 Nonhomogeneous Equations and Undetermined Coefficients (3)

5.6 Forced Oscillations and Resonance (2)

Chapter 6. Eigenvalues and Eigenvectors (3 lectures)

6.1 Introduction to Eigenvalues (1)

6.2 Diagonalization of Matrices (1)

(De-emphasize criteria for diagonalization in terms of subspaces and linear independence)

6.3 Applications Involving Powers of Matrices (1)

Chapter 7. Linear Systems of Differential Equations (8 lectures)

7.1 First-Order Systems and Applications (1)

7.2 Matrices and Linear Systems (3)

7.3 The Eigenvalue Method for Linear Systems (2)

7.4 Second-Order Systems and Mechanical Applications(2)

Chapter 10. Laplace Transform Methods (8 lectures)

10.1 Laplace Transforms and Inverse Transforms (2)

10.2 Transformation of Initial Value Problems (2)

10.3 Translation and Partial Fractions(2)

10.4 Derivatives, Integrals, and Products of Transforms (1)

10.5 Periodic and Piecewise Continuous Input Functions (1)

Examinations, review and leeway (4 lectures)

Total: 55 lectures

Remarks: The linear algebra component of this course should be pursued for 2x2 and 3x3 matrices, even though the exposition is phrased in terms of n x n matrices. Proofs should be given for results having simple proofs, by way of justification, but the emphasis in the course should be on methods.

 

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