Download MA260 - Mohawk Valley Community College

MOHAWK VALLEY COMMUNITY COLLEGE
UTICA, NEW YORK
COURSE OUTLINE
DIFFERENTIAL EQUATIONS
MA260
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Found Acceptable by Gabriel
Found Acceptable by Gabriel
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Found Acceptable by Gabriel
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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:
Course
Objectives:
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)
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:
1.1
1.2
1.3
2)
Determine the type and order of various differential
equations.
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.
Apply the Existence-Uniqueness Theorem to given initial
value problems involving differential equations.
First-Order
Obtainable
Equations
for
which
Exact
Solutions
are
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:
2.1
2.2
2.3
3)
Identify and apply various techniques used to solve exact,
separable, and Bernoulli differential equations.
Solve first-order differential equations with the use of
integrating factors.
Present graphical representation of solutions to selected
differential equations.
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:
3.1
3.2
3.3
4)
Set up and obtain solutions to physical problems from
various disciplines such as chemistry, biology, and physics
(including electricity and mechanics).
Determine and solve differential equations dealing with
motion of a point mass subject to no damping and/or a
damping constant.
Demonstrate problem solving skills by presenting complete,
well organized solutions to application problems involving
differential equations
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:
4.1
4.2
Determine solutions of homogeneous linear differential
equations with constant coefficients.
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:
5.1
5.2
5.3
5.4
5.5
6)
Identify ordinary points for selected differential
equations.
Identify and classify singular points for selected
differential equations.
Solve linear differential equations assuming a solution of
the power series form.
Determine the interval of convergence for an existing power
series solution.
Solve selected differential equations using the method of
Frobenius.
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:
6.1
6.2
6.3
7)
Translate first-order differential equations into
differential operator notation.
Use the methods of elimination and determinants to solve
linear systems of differential equations with constant
coefficients.
Translate and solve application problems dealing with
systems of first and second-order differential equations.
Numerical Solutions to Differential Equations
Various Numerical
Equations
Methods
Applied
to
First
Order
Differential
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:
7.1
7.2
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.
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:
Text:
Chapter 1
1.1
1.2
1.3
2.2
2.3
2.4
2.5
A First Course in Differential Equations, Tenth Edition
by Dennis G. Zill, 2013 Brooks/Cole CENGAGE learning.
Introduction to Differential Equations
5 hours
Definitions and Terminology
Initial – Value Problems
Differential Equations as Mathematical Models
Chapter 2
2.1
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)
First-Order Differential Equations
Solution Curves without a Solution
2.1.1 Direction Fields
2.1.2 Autonomous First-Order DEs
Separable Equations
Linear Equations
Exact Equations
Solutions by Substitutions
12 hours
Note:
2.6
The
emphasis
should
differential equations.
on
solving
homogeneous
A Numerical Method
Note:
Evaluation of students on numerical techniques can be
accomplished
by
assigning
a
computer/calculator
project.
Chapter 3
3.1
3.2
3.3
be
Modeling with First-Order Differential
Equations
4 hours
Linear Models
Nonlinear Models
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
5.1
5.2
5.3
4 hours
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
Linear Models: Boundary-Value Problems (Omit)
Nonlinear Models (Omit)
Chapter 6
6.1
6.2
6.3
6.4
Modeling with Higher-Order Differential
Equations
Series Solutions of Linear Equations
Review of Power Series
Solutions about Ordinary Points
Solutions about Singular Points
Special Functions (Optional)
6 hours
Chapter
7
The Laplace Transform
(Omit)
Chapter 8 Systems of Linear First-Order Differential Equations
(Omit)
Chapter 9
Numerical Solutions of Ordinary Differential
Equations (Optional)
Assessments:
3 hours
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.