## Department of Mathematics - RHIT Topics Outline for MA222 - Differential Equations and Matrix Algebra II - 2010-11

last posted to site: 08/19/10

### Text and Text Materials

Text: Differential Equations with Boundary Value Problems, second edition, by Polking, Bogess, and Arnold

### Approximate Timing and Pace

There should be 2 or 3 tests, perhaps a final review day or two.
It may be advantageous to teach Laplace transforms first. However, no engineering courses have MA222 as corequisite for course require to be taken at the same time as MA222. Therefore, there are no requirements to teach Laplace transforms first.
Optional topics in italics.

 Topics Time Sections Laplace transforms 3 weeks 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7 Systems of differential equations 1 week 1.5 weeks 1 week 3.5 weeks total 8.1, 8.2, 8.3, 8.4, 8.5 9.1, 9.2, 9.3, 9.5, 9.9 10.1, 10.2, 10.5, 10.6, 10.8 Approximation 2.5 weeks 6.1, 11.1, 11.2, 12.1, 12.2, 12.3 Tests and flex/review 1 week

### Laplace Transform Methods - about 3 weeks

Chapter 5: The Laplace Transform
• 5.1 - The Definition of the Laplace Transform.
• 5.2 - Basic Properties of the Laplace Transform
• 5.3 - The Inverse Laplace Transform
• 5.4 - Using the Laplace Transform to Solve Differential Equations.
• 5.5 - Discontinuous Forcing Terms.
• 5.6 - The Delta Function.
• 5.7 - Convolutions

### Systems of differential equations - about 3.5 weeks

Chapter 8:  An Introduction to Systems (about 1 week)

• 8.1 - Definitions and Examples.
• 8.2 - Geometric Interpretation of Solutions.
• 8.3 - Qualitative Analysis.
• 8.4 - Linear Systems.
• 8.5 - Properties of Linear Systems.

Chapter 9:  Linear Systems with Constant Coefficients (about 1.5 weeks)

• 9.1 - Overview of the Technique. Planar Systems. (mostly review of eigenvalues and eigenvectors)
• 9.2 - Planar Systems
• 9.3 - Phase Plane Portraits
• 9.5 - Higher Dimensional Systems (deficient matrices optional)
• 9.9 - Inhomogeneous Linear Systems (undetermined coefficients only)

Chapter 10: Nonlinear Systems (about 1 week)

• 10.1 - The Linearization of a Nonlinear System
(in classification just the characterization: stable, asymptotically stable, unstable, center)
• 10.2 - Long-Term Behavior of Solutions
• 10.5 - Conserved Quantities
• 10.6 - Nonlinear Mechanics (lightly, may need a small amount of material from 10.5)
• 10.8 - Predator - Prey Systems (simple examples)

### Approximation - about 2.5 weeks

Chapter 6: Numerical Methods

• 6.1 - Euler’s Method. (there is one small section on systems)

Chapter 11: Series Solutions to Differential Equations

• 11.1 - Review of Power Series.
• 11.2 - Series Solutions Near Ordinary Points - First and second order equations.

Chapter 12: Fourier Series

• 12.1 - Computation of Fourier Series.
• 12.2 - Convergence of Fourier Series.
• 12.3 - Fourier Cosine and Sine Series.
• Optional topic -Fourier series for solving second order systems with a periodic forcing term - not in text