Chabot College
Fall 2004
Replaced Fall 2010
Course Outline for Mathematics 4
ELEMENTARY DIFFERENTIAL EQUATIONS
Catalog Description:
4 – Elementary Differential Equations
3 units
Introduction to elementary differential equations, including first and second order equations, series
solutions, Laplace transforms, applications. Prerequisite: Mathematics 2 (completed with a grade of C or
higher). 3 hours.
[Typical contact hours: 52.5]
Prerequisite Skills:
Before entry into the course, the student should be able to:
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define the natural logarithmic function in terms of a Riemann integral;
integrate and differentiate logarithmic functions;
define and differentiate inverse functions;
define an exponential function;
differentiate and integrate exponential functions;
differentiate and integrate inverse trigonometric functions;
differentiate and integrate hyperbolic functions and their inverses;
solve application problems involving logarithmic, exponential, inverse trigonometric, and hyperbolic
functions;
solve differential equations using separation of variables;
use standard techniques of integration such as integration by parts, trigonometric integrals,
trigonometric substitution, partial fractions, rational functions of sine and cosine;
graph polar equations and find area of regions enclosed by the graphs of polar equations;
evaluate limits using L’Hopital’s Rule;
evaluate improper integrals;
use parametric representations of plane curves;
perform basic vector algebra in R2 and R3 and interpret the results geometrically;
find equations of lines and planes in R3;
construct polynomial approximations (Taylor polynomials) for various functions and estimate their
accuracy using an appropriate form of the remainder term in Taylor’s formula;
determine convergence of sequences;
determine whether a series converges absolutely, converges conditionally or diverges;
construct (directly or indirectly) power series representations (Taylor’s series) for various functions,
determine their radii of convergence, and use them to approximate function values.
Expected Outcomes for Students:
Upon completion of the course the student should be able to:
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identify certain types of differential equations describing physical problems in sciences and
engineering;
verify that a given solution satisfies a given differential equation and interpret it geometrically when
appropriate;
state the Existence and Uniqueness Theorem (for first order equations only);
identify and solve differential equations in the following categories:
a. first order equations (see Course Content);
b. higher order equations (see Course Content);
c. Laplace Transform;
d. series solutions bear ordinary points;
Chabot College
Course Outline for Math 4, page 2
Fall 2004
5.
e. one of the following topics (instructor’s choice):
1)
series solutions near singular points;
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systems of linear differential equations;
3)
partial differential equations;
solve certain applications:
a. orthogonal trajectories;
b. growth, decay, cooling, circuits, etc.;
c. mechanical vibrations.
Course Content:
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Introduction
a. Classification and some origins of differential equations
b. Geometrical interpretation of equations and solutions
c. Definitions and examples of initial value problems, boundary value problems
d. Existence and Uniqueness Theorem (for first order equations only)
First Order Equations
a. Separable equations
b. Homogeneous equations
c. Exact equations/integrating factors
d. Linear equations
e. Bernoulli equations
f. Equations reducible to first order (substitution)
g. Applications (orthogonal trajectories, growth, decay, cooling, circuits, etc.)
Higher Order Linear Equations with constant coefficients
a. Homogeneous equations
b. Non homogeneous equations by:
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Undetermined coefficients
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Variation of parameters
c. Application of second order linear equations
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Mechanical vibrations (undamped, damped, forced)
The Laplace Transform
Differential Equations with Variable Coefficients
a. Euler Equations
b. Series solutions near ordinary points
Choice of one of the following topics:
a. Series solutions near singular points
b. Systems of linear differential equations
c. Partial differential equations
1)
Separation of variables
2)
The heat or wave equation
3)
Fourier Series
Methods of Presentation:
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Problem-solving
Lecture/discussion
Chabot College
Course Outline for Math 4, page 3
Fall 2004
Assignments and Methods of Evaluating Student Progress:
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Typical Assignments
a. A mass of 1 slug is suspended from a spring whose characteristic spring constant is 9 pound
per foot. Initially the mass starts from a point 1 foot above the equilibrium position with an
upward velocity of 3 feet per second. Find the times for which the mass is heading downward at
a velocity of 3 foot per second.
b. Suppose a function y(t) has the properties that y(0) = 1 and y’(0) = -1. Find the Laplace
transform of y’’ – 4y’ + 5y.
Methods of Evaluating Student Progress
a. Quizzes
b. Exams and final examination
c. Homework
Textbook(s) (typical):
A First Course in Differential Equations, Dennis Zill, PSW Kent Publishers, 2001
Special Student Materials:
A calculator may be required.
CB:al
Revised: 10/20/03