Course Syllabus
Ordinary Differential Equations
SYLLABUS FOR MATH 2320
(Revised January 2009)
Catalog Description: MATH 2320 Ordinary Differential Equations. Topics include
initial value problems for first order and linear second order equations, Picard iterations,
series solutions, and boundary value problems. Laplace transforms and numerical
methods. Prerequisite: MATH 2414.
3 credit (3 lecture).
Prerequisites: MATH 2414.
Course Intent: This course provides the background in sciences for further study in
mathematics and its applications.
Audience: This is a sophomore level mathematics course, which requires a background
consisting of Calculus I and II.
Course Objectives: Upon completion of this course, a student should be able to:
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Verify that a function is a solution for a given differential equation.
Derive a differential equation from a given physical situation.
Determine by inspection at least two solutions of a given initial-value problem.
Solve a given differential equation by separation of variables or by using an appropriate substitution.
Solve given exact differential equations subject in indicated initial conditions.
Solve the given Ricatti equation.
Use Picard’s method to find y1, y2, y3 for a given differential equation.
Find the orthogonal trajectories of a given family of curves.
Determine whether a set of functions are linearly dependent or independent on (-,).
Determine whether an nth-order differential equation is homogeneous, or nonhomogeneous.
Apply the superposition principle for homogeneous and nonhomogeneous equations.
Given a differential equation and one solution, find the second solution.
Solve a given differential equation by undetermined coefficients.
Find a linearly independent function that is annihilated by a given differential operator.
Solve given differential equation by variation of parameters or by involving Cauchy-Euler equations.
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Find the interval of convergence of a given power series.
For a given differential equation, find two linearly independent power series solutions about a point.
Use the method of Frobenius to obtain two linearly independent series solutions about the point x 0=0.
Find the Laplace Transform of a given function.
Find the inverse Laplace Transform of a given function.
Given a Laplace Transform of an integral, evaluate the transform without evaluating the integral.
Use the Laplace transform to solve the given differential equation subject to the given boundaries.
Solve the given system of differential equations by either systemic elimination or determinants.
Use the Laplace transform to solve a given system of differential equations.
Rewrite a given system in normal form.
Solve a given system of equations by either Gaussian elimination or Gauss-Jordan elimination.
Solve a given system of linear first-order equations using matrices.
Solve a given system of homogeneous linear systems.
Use the method of undetermined coefficients to solve a given system on (-,).
Use variation of parameters to solve a given system of equations.
Use matrix exponentials.
Sketch the direction field for a given differential equation.
Given an initial-value problem, use Euler formula to obtain a four-decimal approximation.
Given an initial-value problem, use Runge-Kutta methods to obtain a four decimal approximation.
Use the finite difference method to approximate a solution for a second-order differential equation.
Textbook: A First Course in Differential Equations with Modeling Applications,
Zill, Dennis, Brooks Cole Publishing Company, 7th ed., 2001.
ISBN 0534379990
Course Outline: Instructors may find it preferable to cover the course topics in the order
listed below. However, the instructor may choose to organize topics in any order, but all
material must be covered.
APPROXIMATE TIME
REFERENCE
TEXT
UNIT I - Introduction to Differential Equations
1.2, 1.3
(4 hours)
Sections: 1.1,
This unit begins with some definitions and introduces certain terminology. Included are
Initial-Value problems and some mathematical modeling.
UNIT II - First-order Differential Equations
2.4, 2.5, 2.6
(6 hours)
Sections: 2.1, 2.2, 2.3,
This unit includes the solution of first-order differential equations using the following
techniques/methods: separation of variables, exact equations, linear equations, solutions
by substitution, and a numerical solution.
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UNIT III - Modeling with First-Order Differential Equations
3.2, 3.3
(4 hours)
Sections: 3.1,
This unit solves some of the problems that commonly arise in modeling applications.
UNIT IV - Differential Equations of Higher Order
4.4, 4.5,
(8 hours)
4.8, 4.9
Sections: 4.1, 4.2, 4.3,
4.6, 4.7,
This unit begins with some preliminary theorems including reduction of order. Also
included are the techniques to solve homogeneous linear equations (with constant
coefficients). Cauchy-Euler equations and some nonlinear equations. The chapter
includes the study of Undetermined Coefficients (Superposition approach).
Undetermined Coefficients (Annihilator approach ) and Variation of parameters.
UNIT V - Modeling with Higher Order Differential Equations
5.2, 5.3
(3 hours)
Sections: 5.1,
This unit begins with Linear Equations and initial-value problems. Also, included are
Linear Equations - Boundary-Value problems and nonlinear equations.
UNIT VI -Series Solutions of Linear Equations
6.2, 6.3,
(4 hours)
Sections: 6.1,
This unit begins with a review of Power Series and then proceeds to Power Series
solutions. Also included are solutions about ordinary and singular points. The two
special functions - Bessel’s equation and Legendre’s equation.
UNIT VII -Laplace Transform
7.2, 7.3,
(7 hours)
Sections: 7.1,
7.4, 7.5, 7.6,
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This unit begins with the definition of the Laplace transform with some introductory
examples. Included are the theorems of first and second translation, derivatives of a
transform. Also, transforms of derivatives, integrals and periodic functions and the Dirac
Delta Function. The chapter concludes with systems of Linear Equations.
UNIT VIII -Systems of First-Order Linear Differential Equations
Sections: 8.1, 8.2, 8.3,
(7 hours)
8.4(Optional)
This unit begins with the preliminary theory to solve Homogeneous Linear Systems with
Constant Coefficients (including distinct real eignenvalues, repeated eigenvalues and
complex eigenvalues). Also, included are variation of parameters and matrix exponential.
UNIT IX -Numerical Methods for Ordinary Differential Equations
Sections: 9.1, 9.2, 9.3
(5 hours)
9.4(Optional)
Included in this chapter are Direction Fields, Euler-Methods, Runge-Kutta Method and
Multstep methods.
Departmental Policies:
1. Each instructor must cover all course topics by the end of the semester. The final
exam is comprehensive and questions on it can deal with any of the course objectives.
2. Each student should receive a copy of the instructor’s student syllabus for the course
during the first week of class.
3. A minimum of three in class tests and a comprehensive final examination must be
given. The final examination must be taken by all students.
4. All major tests should be announced at least one week or the equivalent in advance.
5. The final exam must count for at least 25 to 40 percent of the final grade.
6. The final course average will be used in the usual manner (90-100 ”A”; 80-89 “B”;
70-79 “C”; 60-69 “D”; Below 60 “F”).
7. Either an open book or a take home major test may be given at the discretion of the
instructor.
8. Any review sheet should be comprehensive and the student should not feel that
classroom notes, homework, and tests may be ignored in favor of the review sheet for
any examination.
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Resource Materials: Any student enrolled in Math 2320 at HCCS has access to the
Academic Support Center where they may get additional help in understanding the theory
or in improving their skills. The Center is staffed with mathematics faculty and student
assistants, and offers tutorial help, video tapes and computer-assisted drills. Also
available is a student’s Solutions manual which may be obtained from the Bookstore.
Suggested Methods: It is helpful to begin each class with questions concerning the
material discussed and the assigned homework problems. In presenting new material, it
is suggested that an explanation be followed by students working examples in class.
Students should be encouraged to work the review exercises at the end of each chapter.
Also, they should be encouraged to visit the Academic Support Center at their respective
colleges.
Americans with Disabilities Act (ADA)
Any student with a documented disability (e.g. physical, learning, psychiatric,
vision, hearing, etc.) who needs to arrange reasonable accommodations must contact the
Disability Services Office at his or her respective college at the beginning of each
semester. Faculty members are authorized to provide only the accommodations requested
by the Disability Support Services Office. Persons needing accommodations due to a
documented disability should contact the ADA counselor for their college as soon as
possible. Also, interested students may wish to consult the Disability Support Services
Student Handbook which may be found online.
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