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Mathematics and Physics Division
MTH 222 Differential Equations
Course Outline
Course Number & Name: MTH 222 Differential Equations
Credit Hours: 4 .0 Contact Hours: 4.0 Lecture: 4.0 Lab: N/A Other: N/A
Prerequisites: Grade of C or better in MTH 221 or placement
Co-requisites: None Concurrent Courses: None
Course Outline Revision Date: Fall 2010
Course Description: This course covers methods for solving ordinary differential equations together with physical and geometric applications and places particular emphasis on linear equations with constant coefficients. Topics include 1st-order equations, the nth-order linear equation, series solutions, Laplace Transforms, linear systems and numerical methods.
General Education Goals: MTH 222 is affirmed in the following General Education Foundation Category: Quantitative Knowledge and Skills. The corresponding General Education Goal is as follows: Students will use appropriate mathematical and statistical concepts and operations to interpret data and to solve problems.
Course Goals: Upon successful completion of this course, students should be able to do the following:
1. demonstrate knowledge of the fundamental concepts and theories from differential equations;
2. utilize various problem-solving and critical-thinking techniques to set up and solve applied problems in engineering, sciences, business and technology fields;
3. communicate accurate mathematical terminology and notation in written and/or oral form in order to explain strategies to solve problems as well as to interpret found solutions; and
4. use appropriate technology, such as graphing calculators and computer software, effectively as a tool to solve such problems as those described above.
Measurable Course Performance Objectives (MPOs): Upon successful completion of this course, students should specifically be able to do the following:
1. Demonstrate knowledge of the fundamental concepts and theories from differential equations:
1.1 identify the classifications of differential equations and define the solutions of the equations;
1.2 draw the direction field to identify the asymptotical behavior of the solutions;
1.3 derive the general solutions of first order linear equations and solve some nonlinear equations such as exact and separable equations;
1.4 identify the difference between linear and nonlinear equations and apply the Existence and Uniqueness Theorem to determine some qualitative properties of differential equations;
1.5 find the general solutions of second-order and higher-order linear homogeneous equations with constant coefficients by using the characteristic roots;
1.6 use the Undetermined Coefficient Method, and the Variation of Parameters Method to find particular solutions of nonhomogeneous equations;
1.7 use the Power Series Solution Method to solve second-order linear equations by identifying recurrence relations of the coefficients for equations near ordinary or regular singular points;
1.8 define the Laplace Transforms for piecewise continuous functions and use them to find solutions of linear equations, especially equations with piecewise continuous forcing functions;
solve systems of linear equations with constant matrix by using eigenvalues and eigenvectors; and
solve first-order equations numerically by using the Euler method, Picard method and enhanced Euler method
2. Utilize various problem-solving and critical-thinking techniques to set up and solve applied problems in engineering, sciences, business and technology fields:
2.1 set up and solve first-order differential equations modeling the growth and decay and population dynamics applications;
2.2 set up and solve second-order differential equations modeling mechanical and electrical vibrations applications; and
2.3 apply the Laplace transform method on linear equations with step-forcing functions and discontinuous forcing functions arising from engineering applications
3. Communicate accurate mathematical terminology and notation in written and/or oral form in order to explain strategies to solve problems as well as to interpret found solutions:
3.1 write and explain solutions to related rates, optimization, population dynamic and other application problems
4. Use appropriate technology, such as graphing calculators and computer software, effectively as a tool to solve such problems as those described above:
4.1 use a graphing calculator and/or web-based application programs such as Applet to visualize direction fields and asymptotic behavior of solutions of equations; and
4.2 use mathematical software such as Mathematica and Maple to find the power series solutions of equations
Methods of Instruction: Instruction will consist of a combination of lectures, presentation of sample problems, clarification of homework exercises and textbook material, and general class discussion.
Outcomes Assessment: Test and exam questions are blueprinted to course objectives. Data is collected and analyzed to determine the level of student performance on these assessment instruments in regards to meeting course objectives. The results of this data analysis are used to guide necessary pedagogical and/or curricular revisions.
Course Requirements: All students are required to:
1. Maintain regular attendance; excessive absences will negatively affects student understanding and performance.
2. Complete reading and problem-solving homework in a timely manner and contribute to class discussions. Mathematics cannot be understood without doing a significant amount of outside study.
3. Participate in a peer study group that meets regularly and maintains effective member communication links.
4. Take tests and exams when scheduled. No make-ups will be permitted. The first missed test will be recorded as a zero until the end of the semester, at which time the final exam grade will also be used to replace the missing test grade. Grades from any other missed tests will be recorded as irreplaceable zeros. The Comprehensive Final Exam is required and cannot be rescheduled unless some extraordinary event occurs and prior arrangement is made with the instructor.
Methods of Evaluation: Final course grades will be computed as follows:
% of
Grading Components final course grade
Optional Assignments 0 10%
Problem sets, research projects, etc. are designed to enhance understanding of the applications of differential equations in engineering and related fields.
3 or more Tests (dates specified by the instructor) 60 70%
Tests will show evidence of the extent to which students meet course objectives, including, but not limited to, identifying and applying concepts, analyzing and solving problems, estimating and interpreting results, and stating appropriate conclusions using correct terminology.
Final Exam 30 35%
The comprehensive final exam will examine the extent to which students have understood and synthesized all course content and achieved all course objectives.
Note: The instructor will provide specific weights, which lie in the above-given ranges, for each of the grading components at the beginning of the semester. Also, students may use a scientific or graphing calculator or laptop computer to enhance understanding during class or while doing homework. However, no form of technological aid can be used on tests/exams.
Academic Integrity: Dishonesty disrupts the search for truth that is inherent in the learning process and so devalues the purpose and the mission of the College. Academic dishonesty includes, but is not limited to, the following:
plagiarism the failure to acknowledge another writers words or ideas or to give proper credit to sources of information;
cheating knowingly obtaining or giving unauthorized information on any test/exam or any other academic assignment;
interference any interruption of the academic process that prevents others from the proper engagement in learning or teaching; and
fraud any act or instance of willful deceit or trickery.
Violations of academic integrity will be dealt with by imposing appropriate sanctions. Sanctions for acts of academic dishonesty could include the resubmission of an assignment, failure of the test/exam, failure in the course, probation, suspension from the College, and even expulsion from the College.
Student Code of Conduct: All students are expected to conduct themselves as responsible and considerate adults who respect the rights of others. Disruptive behavior will not be tolerated. All students are also expected to attend and be on time all class meetings. No cell phones or similar electronic devices are permitted in class. Please refer to the Essex County College student handbook, Lifeline, for more specific information about the Colleges Code of Conduct and attendance requirements.
Course Content Outline: based on the text Elementary Differential Equations, 9th edition, by William E Boyce & Richard C DiPrima; published by Wiley, Inc., 2008; ISBN #: 9780470623817
Class Meeting
(105minutes) Chapter/Section
Chapter 1 Introduction
1 1.1 Some Basic Mathematical Models; Direction Fields 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations
Chapter 2 First-Order Differential Equations
2 2.1 Linear Equations; Method of Integrating Factors
2.2 Separable Equations 2.3 Modeling with First-Order Equations
3 2.4 Differences Between Linear and Nonlinear Equations 2.5 Autonomous Equations and Population Dynamics
4 2.6 Exact Equations and Integrating Factors 2.7 Numerical Approximations: Euler's Method
5 2.8 The Existence and Uniqueness Theorem
6 Test #1 on Chapters 1 & 2
Chapter 3 Second-Order Linear Equations
7 3.1 Homogeneous Equations with Constant Coefficients
3.2 Fundamental Solutions of Linear Homogeneous Equations; The Wronskian
8 3.3 Complex Roots of the Characteristic Equation 3.4 Repeated Roots; Reduction of Order
9 3.5 Nonhomogeneous Equations; Method of Undetermined
Coefficients 3.6 Variation of Parameters
10 3.7 Mechanical and Electrical Vibrations 3.8 Forced Vibrations
Chapter 4 Higher-Order Linear Equations
11 4.1 General Theory of nth-Order Linear Equations 4.2 Homogeneous Equations with Constant Coefficients
12 4.3 The Method of Undetermined Coefficients 4.4 The Method of Variation of Parameters
13 Test #2 on Chapters 3 & 4
Class Meeting
(105minutes) Chapter/Section
Chapter 5 Series Solutions of Second-Order Linear Equations
14 5.1 Review of Power Series
5.2 Series Solutions Near an Ordinary Point, Part I
15 5.3 Series Solutions Near an Ordinary Point, Part II 5.4 Euler Equations; Regular Singular Points
16 5.5 Series Solutions Near a Regular Singular Point, Part I 5.6 Series Solutions Near a Regular Singular Point, Part II
17 5.7 Bessel's Equation
Chapter 6 The Laplace Transform
18 6.1 Definition of the Laplace Transform 6.2 Solution of Initial Value Problems
19 6.3 Step Functions
20 6.4 Differential Equations with Discontinuous Forcing Functions
21 6.5 Impulse Functions
22 Test #3 on Chapters 5 & 6
Chapter 7 Systems of First-Order Linear Equations
23 7.1 Introduction 7.3 Systems of Linear Algebraic Equations; Linear Independence,
Eigenvalues, Eigenvectors
24 7.4 Basic Theory of Systems of First-Order Linear Equations 7.5 Homogeneous Linear Systems with Constant Coefficients 7.6 Complex Eigenvalues
25 7.7 Fundamental Matrices 7.8 Repeated Eigenvalues
Chapter 8 Numerical Methods
26 8.1 The Euler or Tangent Line Method 8.2 Improvements on the Euler Method 8.3 The Runge-Kutta Method
27 Review for Final Exam
28 Comprehensive Final Exam on all course material covered
page PAGE \* MERGEFORMAT 2prepared by C Wang, Spring 2010
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