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Math 55 --- Differential Equations

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Catalog Description

A study of ordinary differential equations and solutions, equations of first and second order, linear differential equations, systems of equations, phase plane analysis, existence and uniqueness theorems, applications and modeling.

Special notes or advisories: Computer exploration is an integral component of this course. Students will also create and present oral and written analyses of a topic that requires use of the concepts and techniques learned in this course.

Prerequisites

Successful completion of or concurrent enrollment in Math 50B (or equivalent).

Describe representative skills without which the student would be highly unlikely to succeed: Students must be well-grounded in the art of differentiation. Students must also understand the theory of integration and possess basic integration skills.

Recommended Preparation

Math 45 recommended. However, the instructor will cover the linear algebra that students need to be successful in the study of systems of linear differential equations.

Course Learning Outcomes

What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.

1. Read, write, and speak accurately about mathematical ideas and use correct mathematical notation.
2. Use computer technology to solve differential equations and systems numerically, and visualize and interpret their results.
3. Use the theory of differential equations as a fundamental problem-solving tool.
4. Find general and particular solutions of first and second-order linear differential equations, and solve linear systems of two differential equations.
5. Apply the mathematics of differential equations to real-world problems and applications.
6. Use numerical, graphical, symbolic, and verbal representations to solve problems and communicate with others.

Course Content

• Themes: What themes, if any, are threaded throughout the learning experiences in this course?
  • Problem solving
  • Writing
  • Technology
  • Communication
• Concepts: What concepts do students need to understand to demonstrate course outcomes?
  • The use of mathematical software as a fundamental problem-solving tool, particularly in producing numerical solutions of differential equations and systems and in analyzing the results.
  • The presentation of mathematical solutions in a logical coherent structure, including the use of fundamental writing skills, grammar, and punctuation.
  • Qualitative analysis of differential equations and systems.
  • Numerical analysis of differential equations and systems.
  • Symbolic analysis of differential equations and systems.
  • Modeling.
• Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
  • The appropriate use of technology in the problem-solving process.
  • The connection between mathematics and the real world.
  • The role of the student in becoming a successful learner.
  • The recognition that the problem-solving skills learned in this class are applicable in future mathematics classes and classes in related fields, such as physics, engineering, etc.
  • Practical use and limitations of using numerical solvers.
  • Enhancing integration skills.
• Skills: What skills must students master to demonstrate course outcomes?
  1. First-order differential equations:
     • Solve separable and linear first-order equations and initial value problems.
     • Apply differential equations theory to determine if a first-order initial value problem has a unique solution.
     • Use phase line analysis to find equilibrium solutions of autonomous first-order differential equations, determine their stability, and determine the general behavior of other solutions.
     • Derive and solve differential equations that model motion in one direction (including problems involving air resistance), mixtures, population growth, and/or financial scenarios.
  2. Second-order differential equations:
     • Find a fundamental set of solutions to a given second-order linear homogeneous differential equation with constant coefficients.
     • Find the general solution to a given second-order linear inhomogeneous differential equation using the undetermined coefficients method and/or the variation of parameters method.
     • Derive and solve differential equations that model the motion of springs. Also, determine the long-term behavior of the solution.
  3. Numerical methods:
     • Apply Euler and Runge-Kutta methods to approximate the solutions of first-order differential equations.
     • Use mathematical software to approximate the solutions of first-order differential equations and systems of differential equations.
       • Systematically vary the solver options to determine when a solution is accurate enough.
       • Use solver results to determine if a problem is stiff, and vary the solution method accordingly.
  4. Linear systems of differential equations:
     • Use phase plane analysis to visualize the solutions of linear systems of differential equations, and analyze their long-term behavior.
     • Use eigenvalues and eigenvectors to find a fundamental set of solutions to a linear 2x2 system with constant coefficients.
     • Use the eigenvalues and the trace-determinant plane to determine the long-term behavior of solutions.
  5. Nonlinear systems of differential equations:
     • Derive systems of differential equations that model epidemic scenarios and/or predator-prey scenarios.
     • Use phase plane analysis to visualize the solutions of nonlinear systems of differential equations, and analyze their long-term behavior.
     • Find the nullclines and the equilibrium points of a given nonlinear system of differential equations, and then use linearization to classify the equilibrium points and thereby determine the long-term behavior of solutions.

Representative Learning Activities

What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.

• Listening to lectures.
• Participating in group activities and/or assignments.
• Participating in class assignments and/or discussions.
• Completing homework assignments.
• Using mathematical software to complete activities designed to foster a deeper level of understanding of the concepts and skills developed in this class.

Assessment Tasks

How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.

• Representative assessment tasks:
  • Homework assignments.
  • In-class examinations and/or quizzes.
  • Take-home examinations and/or quizzes allow the instructor to include questions and/or exercises that require the use of a computer and mathematical software to supplement the analysis. Extra time allows the students to develop their writing and presentation skills.
  • Writing assignments designed to develop communication of mathematical concepts.
  • Group and/or individual projects and presentations.
• Required assessments for all sections – to include but not limited to:
  • Homework assignments, including computer activities that deepen the level of understanding.
  • Writing assignments designed to develop communication of mathematical concepts.
  • Student projects
  • Examinations.

Examples of Appropriate Texts or Other Readings

• Author: Polking, Boggess, and Arnold; Title: Differential Equations with Boundary Value Problems, Second Edition; Date: 2005
• Author: Polking and Arnold; Title: Ordinary Differential Equations using Matlab, Third Edition; Date: 2004