# 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.