ROCHESTER COMMUNITY & TECHNICAL COLLEGE
COMMON COURSE OUTLINE: Math 2238 - Differential Equations and Linear Algebra

A. Catalog Description

1. Math 2238
2. Differential Equations and Linear Algebra
3. 5 credits
4. 5 hours instruction per week
5. Offered Spring Semester
6. Prerequisites
a. Math 2237
b. College reading level

7. MNTC:
a. Critical Thinking (CT)
b. Mathematics/Symbolic Systems (MA)

8. An in-depth look at topics such as mathematical models, first-order differential equations, applications of linear and nonlinear equations, and other topics.

B. Revised 6/6/2000

C. Recommended entry skills/knowledge – Above average Multivariable Calculus skills
in differentiation and integration, basic understanding of proof and theory, thorough knowledge of handling all types of functions including transcendental, trigonometric, hyperbolic, vector spaces, surfaces with cylindrical and spherical coordinates, derivatives and integrals of vector functions, dot and cross product, double and triple integration and differentiation, plus Power Series and many related topics from Multivariable Calculus.

D. Major Content Areas:

1. Differential Equations –

Definitions and Terminology
Initial-Value Problems
Differential Equations as Mathematical Models
First Order Differential Equations
Modeling with First Order Differential Equations
Linear and Nonlinear Systems of Differential Equations
Higher Order Differential Equations
Initial-Value and Boundary-Value Problems
Homogeneous and Nonhomogeneous Equations
Reductions of Order
Homogeneous Equations with Constant Coefficients
Homogeneous Equations with Undetermined Coefficients using the
Superposition Approach and the Annihilator Approach
Variation of Parameters
Cauchy-Euler Equation
Systems of Linear and Nonlinear Equations, Initial or Boundary-Values
Modeling with Higher-Order Differential Equations
Spring/Mass Systems with Free Undamped Motion
Spring/Mass Systems with Free Damped Motion
Spring/Mass Systems with Driven Motion
Series Solutions of Linear Equations, about Ordinary or Singular Points
Laplace Transforms, Inverse Transforms
Translation Theorems and Derivatives of a Transform
Transforms of Derivatives, Integrals and Periodic Functions
Dirac Delta Function
Systems of Linear First-Order Differential Equations
Homogeneous Linear Systems with Constant Coefficients
Distinct Real Eigenvalues
Repeated Eigenvalues
Complex Eigenvalues
Variation of Parameters
Matrix Exponential

Optional Topics: Direction Fields
Euler Methods
Runge-Kutta Methods
Higher-Order Equations and Systems
Second-Order Boundary-Value Problems

2. Linear Algebra –

Review of Linear Systems
New Applications of Dot Product and Cross Product
Matrix Operations, Elementary and Invertible Matrices
LU Factorization and Applications
Vector Spaces, Subspaces
Linear Independence of Vectors, Bases
Dimension, Coordinate Vectors and Change of Basis
Rank and Nullity
Matrix Transformations
Linear Transformations
Kernel and Range
Matrix and Algebra of Linear Transformations
Determinants, Adjoint, Cramer’s Rule,
Eigenvalues and Eigenvectors, Approximations
Diagonalization
Applications to Dynamical Systems, to Markov Chains
Orthogonal Sets and Matrices,
Orthogonal Projections: Gram-Schmidt Process
QR Factorization
Least Squares Method
Orthogonalization of Symmetric Matrices
Quadratic Forms and Conic Sections

Optional Topics:
Singular Value Decomposition
Inner Products
Additional Applications of the topics listed
Computer Projects using Mathematica Software

E. Learning Outcomes

1. Mathematical/Logical Reasoning from MN Transfer Curriculum (MTC)
a. Illustrate historical and contemporary applications of mathematical/logical reasoning.
b. Clearly express mathematical/logical ideas in writing.
c. Explain what constitutes a valid mathematical/logical argument (proof).
d. Recognize higher order problem-solving and/or modeling strategies.

1. Critical Thinking from MN transfer curriculum
a. Gather factual information and apply it to a given problem in a manner that is relevant, clear, comprehensive, and conscious of possible bias in the information selected.
b. Imagine and seek out a variety of possible goals, assumptions, interpretations, or perspectives which can give alternative meanings or solutions to given situations or problems.

c. Analyze the logical connections among the facts, goals, and implicit assumptions relevant to a problem or claim; generate and evaluate implications that follow from them.
d. Recognize and articulate the value assumptions which underlie and affect decisions, interpretations, analyses, and evaluations made by ourselves and others.

Other Competencies:

Need to have skilled use of the graphics calculator.
Must acquire some skills in a computer software application, e.g. Mathematica.
Must have college reading level for reading the text and interpreting the problems.
Should have above average skills in Multivariable Calculus integration and
differentiation. Be able to learn and use precise thinking and writing methods for structuring
problems.
a. Mastery of Major Content Areas in Part D.

b. Introduction of Applications of these concepts to the degree that time allows.

F. Methods used for Evaluation
1. Test over the covered topics for each chapter
2. Quizzes and Extra Credit Opportunities
3. Homework
4. Group Assignments
5. Final Exam from the untested topics using skills and methods mastered in this class.

G. Graphing Calculators needed and used in this class.