# 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.