## Rose-Hulman - Department of Mathematics - Course Syllabus MA222 - Differential Equations and Matrix Algebra II - 2010-11

 Description & Prerequisites Course Goals Texts and Other Materials Course Topics Course Policies Links last posted to site: 08/19/10

Parts of the web page to be completed or determined by the instructor are in green.

### Catalogue Description and Prerequisites

MA 222 Differential Equations and Matrix Algebra II 4R-0L-4C W, S Pre: MA 221
Solution of systems of first order linear differential equations by eigensystems and investigation of their solution structure determined by eigensystems. Phase portrait analysis and classification of the nature of the stability of critical points for linear and nonlinear systems. Laplace transforms. Solving small systems of first order linear differential equations by Laplace transforms. Series solutions. Fourier series. Applications to problems in science and engineering.

Prerequisite: All of the topics in MA221 will be assumed. In addition, familiarity with some Maple commands form Calculus is assumed. The Maple files  MapleIntRGL.mws (introduction) and calcrev.mws (review for DE) may be helpful. These files are available on Angel in the (Rose-Only) Mathematics Course Information Repository.

### Course Goals

1. Provide an introduction to systems of differential equations.
2. Provide an introduction to Laplace transform methods.
3. Provide an introduction to Fourier series and series methods of solutions.
4. Develop a deeper understanding and appreciation of transformation and approximation methods (by studying Laplace, series, and Fourier methods).
5. Improve mathematical modeling and analytical problem solving skills.
6. Develop ability to communicate mathematically.
7. Improve skill using the computer as a tool for mathematical analysis and problem solving.
8. Introduce applications of mathematics, especially to science and engineering.

### Textbook and other required materials

Textbook: Text: Differential Equations with Boundary Value Problems, second edition, by Polking, Bogess, and Arnold
Computer Usage:  Maple13 must be available on your laptop.

Course Topics

1. Laplace Transforms
• Definition of Laplace transform and standard table; inverse transform
• Basic properties and theorems
• Delta and Heaviside functions
• Convolution
• Transfer functions
• Solution of linear systems by Laplace transforms - briefly by examining small systems only
• Applications as appropriate - e.g., salt tanks, spring-mass systems, and electrical circuits.
2. Systems of first order differential equations
• Solution and solution structure determined by eigensystem, for x' = Ax and x' = Ax + b, (b constant and A invertible)
• Trajectories and phase portrait for x' = Ax and x' = Ax + b near critical points
• Classification and stability of critical points for linear systems
• Phase portrait, linearization, and stability of critical points for non-linear systems
• Numerical solutions of systems
• Applications as appropriate - e.g., predator/prey and competing species for phase portraits, tanks for eigenvalue methods
3. Approximation Fourier Series
• Numerical solutions
• Sine and cosine series
• Applications to ordinary differential equations, as appropriate - e.g., endpoint problems, steady-state solutions with periodic forcing functions

### Course Requirements and Policies

#### Computer Policy

A summary of the computer policy page:

Students will be expected to demonstrate a minimal level of competency with a relevant computer algebra system. The computer algebra system will be an integral part of the course and will be used regularly in class work, in homework assignments and during quizzes/exams. Students will also be expected to demonstrate the ability to perform certain elementary computations by hand. (See Performance Standards below.)

#### Performance Standards

##### Paper and pencil and Laplace transform table
1. Solve 2x2 systems of differential equations (linear, autonomous, including x' = Ax + b)
2. Compute elementary Laplace transforms from the definition and standard table.
3. Solve simple scalar differential equations using Laplace transforms
4. Linearize systems of non-linear equations, determining and classifying critical points
5. Sketch  phase portraits of simple systems
6. Compute Fourier coefficients of simple functions
7. Euler's method for two variable systems - a couple of iterations
8. Solve simple ODE's using power series.
##### Maple
1. Maple competencies from the calculus sequence and MA221
2. Use Maple to solve systems of differential equations, including plotting solutions and solution curve
3. Use Maple to plot and analyze phase portraits and direction fields, including critical point analysis
4. Use Maple to compute Laplace and inverse Laplace transforms
5. Use Maple to solve and analyze differential equations, using Laplace transforms
6. Use Maple to compute Fourier series

#### Final Exam Policy

The following is an extract from the final exam policy page. Consult the policy page for complete details.

The final exam will consist of two parts. The first part will be "by hands" (paper, pencil). No computing devices (calculators/computers) will be allowed during the first part of the final exam. This part of the exam will cover both computational fundamentals as well as some conceptual interpretation, though the level of difficulty and depth of conceptual interpretation must take into account that this part of the exam will be shorter than the second part of the exam.  The laptop, starting with a blank Maple work sheet, and a calculator, may be used during the second part of the exam. No "cheat sheets", prepared Maple worksheets or prepared program on the calculator may be used. The second part of the exam will cover all skills: concepts, calculation, modeling, problem solving, and interpretation.

#### Individual Instructor Policies

• the grading scheme, based on the various course components.
• the number and format of hour exams, quizzes, homework assignments, in class assignments, and projects,
• the policies governing the work items above, e.g., whether the computer will be used, what collaboration is allowed, and the format of assignments.
• all policies for classroom procedure, including group work, class participation, laptop use and attendance*.
*Note that most instructors will enforce some type of grade penalty for students with more than four unexcused absences.