# MATH-260 Differential Equations

**Description**

MATH-260 consists of concepts generally encountered in a
first course in differential equations . This includes a

comprehensive treatment of first order differential equations employing a
variety of solution techniques . A study

of higher order equations, largely second order, is included with emphasis on
linear equations possessing

constant coefficients as well as variable coefficients. Classical and
contemporary applications are included

throughout that come from diverse fields such as mechanics, electrical circuits,
economics, and possibly from

areas of special student interest. Computer uses with MATHLAB software provide
an integrated environment

for symbolic, graphic, and numeric investigations of routine solutions of
differential equations and of modeling

physical phenomena. The course concludes with a discussion of Laplace transforms
and systems of linear

equations.

**Prerequisites:** Completion of a calculus sequence,
equivalent to MATH -182. A grade of C or better is strongly

recommended.

**Statement on General Education and Liberal Learning**

A liberal education prepares students to lead ethical,
productive, and creative lives and to understand how

the pursuit of lifelong learning and critical thinking fosters good citizenship.
General education courses

form the core of a liberal education within the higher education curriculum and
provide a coherent

intellectual experience for all students by introducing the fundamental concepts
and methods of inquiry in

the areas of mathematics, the physical and natural sciences, the social
sciences, the arts and the humanities,

and composition. This course is part of the general education core experience at
Howard Community

College.

**Objectives:** The general objective of MATH-260 is to
develop the basic ideas commonly encountered in a first

course in differential equations, to demonstrate some of their many
applications, to enhance computer/calculator

literacy, and to promote mathematical maturity for more advanced studies in
mathematics. Successful

completion of MATH-260 can be briefly described by the acquisition of the
following behaviors.

State and use basic definitions and theorems, correctly
use standard symbolism,

and accurately and quickly perform required computations both manually and

with the support of MATLAB software.

Build, solve and analyze mathematical models.

Translate the basic ideas of ordinary differential equations between their
analytic

and their graphic representations

Solve routine application problems for first and second order ordinary
differential

equations

Solve simple non-routine problems so as to extend the scope of a topic to solve

problems amid slightly altered conditions

Follow mathematical reasoning as provided in elementary proofs, develop logical

arguments, and identify mathematical patterns.

**General Approach with MATLAB:** The general approach
of the course falls under the following themes:

1. Existence and uniqueness of solutions

2. Dependence of solutions on initial values.

3. Derivation of formulas for solutions .

4. Numerical calculation of solutions .

5. Graphical analysis of solutions.

6. Qualitative analysis of differential equations and their solutions.

The symbolic, numerical, and graphical capabilities of
MATLAB will be used to analyze differential equations

and their solutions.

**Major Topics**

**First Order Differential Equations**

- Linear Equations with Variable Coefficients

- Separable Equations

- Modeling with First Order Equations

- Difference Between Linear and Non-linear Equations

- Exact Equations and Integrating Factors

- Numerical Approximation: Euler’s Method

- Existence and Uniqueness Theorem

**Second Order and Higher order Linear Equations**

- Homogeneous Equations with Constant Coefficients

- Fundamental Solutions of Linear Homogeneous Equations

- Linear Independence and The Wronskian

- Complex Roots of the Characteristic Equations

- Repeated Roots: Reduction of Order

- Non-homogeneous Equations; Method of Undetermined Coefficients

- Variation of Parameters

- Mechanical Vibrations and Electrical Oscilations

- Forced Vibrations

**Series Solutions of Second Order Linear Equations**

- Review of Power Series

- Series Solution near an Ordinary Point I and II

- Euler Equations

**The Laplace Transform**

- Definition of the Laplace Transform

- Solution of Initial Value Problems

- Step Functions

- Differential Equations with Discontinuous Forcing Functions

- Impulse Functions (Optional)

**Systems of linear differential Equations**

- Application Laplace transforms to systems of differential equations or the use
of the operator method

Prev | Next |