SANTA BARBARA CITY COLLEGE
ASSOCIATE
DEGREE CREDIT COURSE OUTLINE
Department: Mathematics
Subject Area and Course Number: Mathematics 220
Course Title:
Differential Equations
Discipline: Mathematics
Units: 4
Repeatability: None
Catalog Course Description: Introductory course in the theory and
applications of ordinary and partial differential equations. Topics include
constant coefficient equations, series techniques, introduction to Laplace
Transforms, qualitative and quantitative solutions to linear and nonlinear
systems of differential equations, and separable partial differential
equations.
Description for Schedule of Classes:
Ordinary and partial differential equations, with applications, series
techniques, Laplace Transforms, qualitative and quantitative solutions to
linear and nonlinear systems of differential equations.
Lecture Hours per Week: 4.3 (64-72
Total Semester Hours)
Laboratory Hours per Week: None
Plus Hours: None
Prerequisites: Math 200, with a grade of "C"
or better.
Co-requisites: None
Skills Advisories: None
Course Advisories: None
Limitation on Enrollment: None
Course Objectives: By the end of this course, the student
will be able to:
1.
Solve linear, exact, and separable
ordinary and separable partial differential equations.
2.
Write differential equations to
represent some types of natural phenomena.
3.
Use transform techniques in problems
involving discontinuous and impulsive forcing functions.
4.
Apply matrix techniques to solve
systems of linear differential equations.
5.
Solve differential equations using
series techniques.
6.
Analyze critical points and
stability for systems of equations.
Course Content and Scope
1. First
Order Differential Equations
a. Existence
and Uniqueness Theorems
b. Solution
of linear equations
c. Separable
Equations
d. Exact
Equations
e. Applications
2. Second
Order Linear Equations
a. Homogeneous
equations
b. Linear
independence of solutions (Wronskian)
c. Nonhomogeneous
equations
d. Applications
3. Series
Solutions–Variable coefficients
a. Ordinary
points
b. Regular
singular points
4. Laplace
transforms
a. Initial
value problems
b. Step
functions
c. Impulse
functions
d. Convolutions
5. Systems
of first order linear equations
a. Eigenvalue–Eigenvector
method for solving homogeneous systems with constant coefficients
b. Fundamental
matrices and matrix exponential
c. Nonhomogeneous
systems
6. Partial
Differential Equations
a. Fourier
Series
b. Separable
equations
c. The
heat equation
d. The
wave equation
e. Laplace's
equation
7. Numerical
Methods
a. Euler's
Method
b. Runge-Kutta
Method
8. Nonlinear
Differential Equations and Systems
a. Stability
and the Phase Plane
b. Linear
and almost linear systems
Methods of Instruction: A combination of lecture and computer
explorations will be used in the course. Students will be exposed to software
and graphical approaches to the subject as well as to material from texts.
Required Assignments
1. Appropriate
Readings: Students are required to
read assigned chapters in texts.
2. Writing
Assignments: Students must work
assigned mathematical problems requiring the understanding of abstract ideas.
3. Appropriate
Outside Assignments: Students will
be expected to spend a sufficient amount of time outside of class to practice
techniques taught during class time, read assigned materials, and complete
frequent homework and computer assignments.
4. Appropriate
Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve
analyzing information, recognizing concepts in new contexts, and drawing
analogies. They must also analyze logical arguments for validity and write
proofs of their own using both inductive and deductive reasoning within a
logical system. Students will also learn to use software tools in solving differential
equations.
Methods of Evaluation: A student's grade will be based on
multiple measures of performance in the solving of problems, designing of
mathematical models, preparation and analysis of graphs, and analysis of
logical arguments. Such measures
will include at least four one-hour exams and a comprehensive final examination
requiring demonstrations of problem solving skills. In addition, instructors may make use of quizzes, written
homework assignments, computer assignments, or other appropriate means to judge
a student's dexterity with mathematical skills and familiarity with
mathematical vocabulary and methods of proof.
Appropriate Texts and Supplies
Zill and Cullen, Differential Equations with Boundary-Value Problems, 7th Ed., Cengage Publishing, 2009
Maple, Mathematica, Matlab, or equivalent computer algebra system
Student Learning Outcomes:
1.
Apply
Differential Equations to problems in the sciences.
2.
Solve
various linear and nonlinear ODEÕs analytically or numerically.
3.
Determine
the qualitative behavior of an autonomous nonlinear system by means of an
analysis of behavior near critical points.
4.
Use
Laplace transforms to solve second order linear ODEÕs with discontinuous
forcing functions or impulse functions.
5.
Compute
Fourier coefficients, and find periodic solutions of linear ODE's and PDE's by
means of Fourier series and separation of variables.
JK/mej
Rev
9/24/07; 8/24/09
FRC
(WPC)