ModelCourses/Ordinary Differential Equations

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General Description

• Sophomore or junior level course
• Pre-requisite: Calculus

Possible textbooks include, but are not limited to:

• Fundamentals of Differential Equations and Boundary Value Problems, by Nagle, Saff, and Snider, Addison-Wesley.
• Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima, Wiley.

Course Objectives

Upon successful completion of this course, students will be able to:

• Determine if a given function is a solution to a particular differential equation; apply the theorems for existence and uniqueness of solutions to differential equations appropriately;
• Distinguish between
  (a) linear and non-linear differential equations;
  (b) ordinary and partial differential equations;
  (c) homogeneous and non-homogeneous differential equations;
• Solve ordinary differential equations and systems of differential equations using:
  (a) Direct integration
  (b) Separation of variables
  (c) Reduction of order
  (d) Methods of undetermined coefficients and variation of parameters
  (e) Laplace transform methods
• Determine particular solutions to differential equations with given initial conditions.
• Analyze real-world problems such as motion of a falling body, compartmental analysis, free and forced vibrations, etc.; use analytic technique to develop a mathematical model, solve the mathematical model and interpret the mathematical results back into the context of the original problem.
• Apply matrix techniques to solve systems of linear ordinary differential equations with constant coefficients.
• Find the general solution for a first order, linear, constant coefficient, homogeneous system of differential equations; sketch and interpret phase plane diagrams for systems of differential equations.

Problem Sets

Use of Problem Sets

The problem sets were assembled to allow for personalization by individual faculty. The topics covered are fairly standard in an introductory Differential Equations course, but faculty can rearrange the topics and delete any sections they do not wish to cover, or wish to assess by other means. The names of the problem sets are meant to be descriptive and the learning objectives will help you evaluate if the set should be included or not.

Download the problem sets

A copy of the course can be found at at the MAA website
The course can be downloaded here <insert link>.

To use the files remove the .txt from the end. The .tgz can be added. This file can now be directly uploaded into your own course:

• go to Filemanager
• Upload the file
• etc <provide enough detail to allow for easy installation by anyone>

Description of Problem Sets

• Chapter 0 : Introduction
  • 01 : Solutions and Initial Value Problems
    Students will be able to:
    • Determine if a given function is a solution to a particular differential equation
    • Identify elements of a given solution to a differential equation
    • Model with differential equations
  • 02 : Classification of Differential Equations
    Students will be able to distinguish between:
    • linear and non-linear differential equations;
    • ordinary and partial differential equations;
    • homogeneous and non-homogeneous higher order differential equations

• Chapter 1 : First-Order Equations
  • 01 : Integrals as Solutions
    Students will be able to:
    • Solve simple first order differential equations by direct integration
    • Solve simple differential equations using their knowledge from Calculus
  • 02 : Existence and Uniqueness Theorem
    
  • 03 : Slope Fields and Equilibrium Solutions
    
  • 04 : Separable Equations
    
  • 05 : Linear Integrating Factors
    
  • 06 : Exact Equations
    
  • 07 : Substitution Techniques and Integrating Factors
    
  • 08 : Autonomous
    
  • 09 : Euler's Method
    
  • 10 : Applications
    
  • 11 : Miscellaneous
    

• Chapter 2 : Higher-Order Equations
  • 01 : Introduction to Linear
    
  • 02 : Linear 2nd-Order Constant-Coefficient Homogeneous Equations
    
  • 03 : Linear Higher-Order Constant-Coefficient Homogeneous Equations
    
  • 04 : Free Mechanical Vibrations
    
  • 05 : Nonhomogeneous Undetermined Coefficients
    
  • 06 : Nonhomogeneous Variation of Parameters
    
  • 07 : Forcing Resonance
    

• Chapter 3 : Linear Systems
  • 01 : Intro to Systems
    
  • 02 : Matrices
    
  • 03 : Linear Systems
    
  • 04 : Eigenvalue Method
    
  • 05 : 2D Systems Vector Fields
    
  • 06 : Second Order Systems Mass Spring Systems
    
  • 07 : Repeated Eigenvalues
    
  • 08 : Complex Eigenvalues
    
  • 09 : Nonhomogeneous Systems
    

• Chapter 4 : Laplace Transforms
  • 01 : Laplace-transforms
    
  • 02 : Shifts-and-IVPs
    
  • 03 : Partial-fractions
    
  • 04 : Periodic-functions
    
  • 06 : Convolution
    
  • 07 :Delta-function
    

WeBWorK Workshop, Ann Arbor Michigan June 2013