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MOHAWK VALLEY COMMUNITY COLLEGE

UTICA, NEW YORK

COURSE OUTLINE

DIFFERENTIAL EQUATIONS

MA260

Reviewed and Found Acceptable by Norayne Rosero - 5/01

Reviewed and Revised by Norayne Rosero – 1/02

Reviewed and Revised by Norayne Rosero – 5/02

Reviewed and Revised by Norayne Rosero – 5/03

Reviewed and Revised by Norayne Rosero – 5/04

Reviewed and Revised by Norayne Rosero – 5/05

Reviewed and Revised by Norayne Rosero – 5/06

Reviewed and Revised by Norayne Rosero – 10/07

Reviewed and Found Acceptable by Norayne Rosero – 5/08

Reviewed and Revised by Norayne Rosero – 11/08

Reviewed and Revised by Norayne Rosero – 5/09

Reviewed and Found Acceptable by Norayne Rosero – 5/10

Reviewed and Found Acceptable by Gabriel Melendez – 5/11

Reviewed and Found Acceptable by Gabriel Melendez – 5/12

Reviewed and Found Acceptable by Gabriel Melendez – 5/13

Reviewed and Revised by Gabriel Melendez – 10/13

Reviewed and Found Acceptable by Gabriel Melendez – 5/14

Reviewed and Found Acceptable by Gabriel Melendez – 5/16

Reviewed and Found Acceptable by Gary Gulis-6/18

Course Outline

Title: Differential Equations

Catalog No. MA260

Credit Hours: 3

Lab Hours: 0

Prerequisites: MA152 Calculus 2

NOTE: While there is no physics prerequisite for this course, it is recommended that the instructor warn the student that some problems will be chosen from areas involving concepts of physics. The student who has never had a course in these areas should realize that he/she will have to do extra work in order to be successful.

Catalog

Description: This course introduces the concepts and theory of ordinary differential equations. Topics include existence and uniqueness of solutions, and separable, homogenous, exact, and linear differential equations. Methods involving integrating factors, undetermined coefficients, variation of parameters, power series, numerical approximation, and systems of differential equations using differential operators are covered. Applications are drawn from geometry, chemistry, biology, and physics. Prerequisite MA152 Calculus 2 (Spring Semester only)

Course

Objectives: 1) To acquaint the student with the properties of ordinary differential equations and their solutions

2) To acquaint the student with existing techniques for solving the more commonly occurring differential equations

3) To develop the student's ability to analyze a physical problem involving differential equations.

SUNY Learning Outcomes

1. The student will develop well reasoned arguments by demonstrating an ability to write proofs.

2. The student will identify, analyze, and evaluate arguments as they occur in their own and other’s work.

3. The student will demonstrate the ability to interpret and draw inferences from mathematical models such as formulas, graphs, tables, and schematics.

4. The student will demonstrate the ability to represent mathematical information symbolically, visually, numerically, and verbally.

5. The student will demonstrate the ability to employ quantitative methods such as arithmetic, algebra, geometry, or statistics to solve problems.

6. The student will demonstrate the ability to estimate and check mathematical results for reasonableness.

Major Topics:

1) Differential Equations and Their Solutions

Classification of Differential Equations; Their Origin and Application; Solutions; Initial-Value Problems; Boundary-Value Problems and Existence of Solutions

Topic Goal:

To help the student classify and identify properties of selected ordinary differential equation and their solutions and applications.

Student Outcomes:

The student will:

1. Determine the type and order of various differential equations.

2. Demonstrate an understanding of concepts dealing with solutions of differential equations (explicit and implicit and general and particular solutions) and of initial and boundary value problems involving differential equations.

3. Apply the Existence-Uniqueness Theorem to given initial value problems involving differential equations.

2) First-Order Equations for which Exact Solutions are Obtainable

Exact Differential Equations and Integrating Factors; Separable Equations and Equations Reducible to This Form; Linear Equations and Bernoulli Equations; Integrating Factors

Topic Goal:

To help the student apply various techniques to systematically solve differential equations through the use of numerical, analytical and/or graphical methods.

Student Outcomes:

The student will:

1. Identify and apply various techniques used to solve exact, separable, and Bernoulli differential equations.

2. Solve first-order differential equations with the use of integrating factors.

3. Present graphical representation of solutions to selected differential equations.

3) Applications of First Order Differential Equations

Applied problems are drawn from geometry, chemistry, biology and physics. Emphasis is placed on the mathematical modeling of the physical system as well as on the correct application of the methods of solutions learned.

Topic Goal:

To help the student analyze a real-world physical application problem, create a reasonable mathematical model of a system using a differential equation and solve the resulting equation.

Student Outcomes:

The student will:

1. Set up and obtain solutions to physical problems from various disciplines such as chemistry, biology, and physics (including electricity and mechanics).

2. Determine and solve differential equations dealing with motion of a point mass subject to no damping and/or a damping constant.

3. Demonstrate problem solving skills by presenting complete, well organized solutions to application problems involving differential equations

4) Explicit Methods of Solving Higher-Order Linear Differential

Equations

Basic Theory of Linear Differential Equations; Homogeneous Linear Equations with Constant Coefficients; The Method of Undetermined Coefficients; Variation of Parameters

Topic Goal:

To help the student apply various techniques to systematically solve higher order differential equations.

Student Outcomes:

The student will:

1. Determine solutions of homogeneous linear differential equations with constant coefficients.

2. Demonstrate problem solving skills by presenting complete solutions to non-homogeneous differential equations using the methods of reduction of order, variation of parameters, and undetermined coefficients.

5) Series Solutions to Linear Differential Equations

Ordinary and Singular Points; Obtaining Linearly Independent Solutions; Recurrsion Relationships, Method of Frobenius

Topic Goal:

To help student use power series to obtain the solution of given initial value problems involving differential equations.

Student Outcomes:

The student will:

1. Identify ordinary points for selected differential equations.

2. Identify and classify singular points for selected differential equations.

3. Solve linear differential equations assuming a solution of the power series form.

4. Determine the interval of convergence for an existing power series solution.

5. Solve selected differential equations using the method of Frobenius.

6) Systems of Linear Differential Equations

Homogeneous and Nonhomogeneous Systems

Operator Method of Solution of Systems

Topic Goal:

To help students learn to solve systems of equations involving differential equations.

Student Outcomes:

The student will:

1. Translate first-order differential equations into differential operator notation.

2. Use the methods of elimination and determinants to solve linear systems of differential equations with constant coefficients.

3. Translate and solve application problems dealing with systems of first and second-order differential equations.

7) Numerical Solutions to Differential Equations

Various Numerical Methods Applied to First Order Differential Equations

Topic Goal:

To help students learn to use appropriate technology to apply numerical methods in obtaining the solution to selected differential equations.

Student Outcomes:

The student will:

1. Use a computer or graphing calculator to apply numerical methods (Euler Method, Improved Euler Method, Runge-Kutta Method) to approximate solutions of first-order initial value problems involving differential equations.

2. Compare the accuracy of the answer obtained by numerical methods to the exact answer of a given differential equation.

Teaching Guide

Title: Differential Equations

Catalog Number: MA260

Credit Hours: 3

Lab Hours: 0

Prerequisites: MA152 Calculus 2

NOTE: While there is no physics prerequisite for this course, it is recommended that the instructor inform the student that some problems will be chosen from areas involving concepts of physics. The student who has never had a course in these areas should realize that he/she will have to do extra work in order to be successful.

Catalog

Description: This course introduces the concepts and theory of ordinary differential equations. Topics include existence and uniqueness of solutions, and separable, homogenous, exact, and linear differential equations. Methods involving integrating factors, undetermined coefficients, variation of parameters, power series, numerical approximation, and systems of differential equations using differential operators are covered. Applications are drawn from geometry, chemistry, biology, and physics. Prerequisite MA152 Calculus 2 (Spring Semester only)

Text: A First Course in Differential Equations, Tenth Edition by Dennis G. Zill, 2013 Brooks/Cole CENGAGE learning.

Chapter 1 Introduction to Differential Equations 5 hours

1.1 Definitions and Terminology

1.2 Initial – Value Problems

1.3 Differential Equations as Mathematical Models

Chapter 2 First-Order Differential Equations 12 hours

1. Solution Curves without a Solution

1. Direction Fields

2. Autonomous First-Order DEs

2. Separable Equations

3. Linear Equations

4. Exact Equations

5. Solutions by Substitutions

Note: The emphasis should be on solving homogeneous differential equations.

6. A Numerical Method

Note: Evaluation of students on numerical techniques can be accomplished by assigning a computer/calculator project.

Chapter 3 Modeling with First-Order Differential 4 hours

Equations

1. Linear Models

2. Nonlinear Models

3. Modeling with Systems of First-Order DEs

Chapter 4 Higher-Order Differential Equations 11 hours

4.1 Preliminary Theory – Linear Equations

4.1.1 Initial-Value and Boundary-Value Problems

4.1.2 Homogeneous Equations

4.1.3 Nonhomogeneous Equations

4.2 Reduction of Order

4.3 Homogeneous Linear Equations with Constant Coefficients:

4.4 Undetermined Coefficients – Superposition Approach

4.5 Undetermined Coefficients – Annihilator Approach (Optional)

4.6 Variation of Parameters

4.7 Cauchy-Euler Equation

4.8 Green’s Function (Omit)

4.8.1 Initial-Value Problems

4.8.2 Boundary-Value Problems

4.9 Solving Systems of Linear Equations by Elimination

4.10 Nonlinear Differential Equations (Omit)

Chapter 5 Modeling with Higher-Order Differential 4 hours

Equations

5.1 Linear Models: Initial-Value Problems

5.1.1 Spring/Mass Systems: Free Undamped Motion

5.1.2 Spring/Mass Systems: Free Damped Motion

5.1.3 Spring/Mass Systems: Driven Motion

5.1.4 Series Circuit Analogue

5.2 Linear Models: Boundary-Value Problems (Omit)

5.3 Nonlinear Models (Omit)

Chapter 6 Series Solutions of Linear Equations 6 hours

6.1 Review of Power Series

6.2 Solutions about Ordinary Points

6.3 Solutions about Singular Points

6.4 Special Functions (Optional)

Chapter 7 The Laplace Transform (Omit)

Chapter 8 Systems of Linear First-Order Differential Equations (Omit)

Chapter 9 Numerical Solutions of Ordinary Differential 3 hours

Equations (Optional)

Assessments: The teaching guide allows 3 additional hours for the in-class assessment of student learning. A two hour comprehensive final examination will also be given.

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