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Introduction to Ordinary Differential Equations

MTH 264

Introduction to Ordinary Differential Equations

MTH 264

Course Description

Prerequisite: MTH 261 with a grade of "C" or better. Studies the techniques for solving first and second-order differential equations and first-order systems of differential equations both linear and nonlinear, through qualitative, quantitative and numerical approaches. Includes Laplace transforms and uses applications in science and engineering throughout the course. (45-0)

Outcomes and Objectives

Develop the ability to recognize, classify, and solve different types of first-order differential equations.

Objectives:

  • Identify and solve separable and linear first-order equations.
  • Use slope fields and equilibrium solutions to understand the qualitative properties of first-order equations.
  • Use the concept of bifurcation to understand the qualitative properties of a family of first-order equations.
  • Apply numerical methods to generate approximations to solutions of first-order equations.
  • Understand the conditions that guarantee the existence and uniqueness of solutions to first-order equations.

Solve first-order systems of differential equations and demonstrated knowledge of properties and applications.

Objectives:

  • Use direction fields and equilibrium solutions to understand the qualitative properties of first-order systems.
  • Solve decoupled and partially-decoupled systems.
  • Apply numerical methods to generate approximations of solutions of first-order systems.
  • Investigate the special properties of linear systems.
  • Classify and solve first-order linear systems with constant coefficients.
  • Use first-order linear systems to investigate the properties and solve equations arising from harmonic oscillation.
  • Linearize nonlinear systems when appropriate.
  • Apply appropriate quantitative, qualitative, and numerical techniques to study nonlinear systems.

Analyze and solve second-order differential equations and use them to various applications.

Objectives:

  • Identify homogeneous and nonhomogeneous linear differential equations.
  • Construct particular and general solutions to homogeneous linear differential equations.
  • Construct particular and general solutions to linear differential equations.
  • Solve linear differential equations with constant coefficients.
  • Use second-order linear differential equations to model damped/undamped forced/unforced oscillations.
  • Apply power series to solve or approximate solutions of differential equations.

Use Laplace transforms to solve a variety of differential equations.

Objectives:

  • Apply the definition and properties of the Laplace transform.
  • Apply Laplace transforms to various fundamental functions.
  • Apply the shifting theorems to a variety of functions and equations.
  • Use Laplace transforms to solve a variety of initial value problems.
  • Understand and use the Laplace transform in applications of discontinuous forcing functions.
  • Use the convolution theorem on appropriate first- and second-order equations.

Use appropriate technology to investigate and solve differential equations.

Objectives:

  • Generate and graph numerical solutions with a computer algebra system.
  • Graph and recognize the relationships between forcing functions and solutions to harmonic oscillation.
  • Recognize initial conditions in the graphs of solutions to first-order equations and systems.
  • Generate and graph slope fields and direction fields for first-order equations and systems.
  • Recognize and verify the correspondence between slope/direction fields and solutions of equations or systems.
  • Graph multiple representations of solutions to first-order systems.

Communicate effectively about differential equations and their applications.

Objectives:

  • Verbally describe solutions to problems using appropriate terminology.
  • Provide complete written solutions to problems using appropriate terminology.
  • Use appropriate vocabulary.