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Aktosun, Spring 2017, Math 3319

**
Instructor:** Tuncay Aktosun, PKH 461, (817) 272-1545,
aktosun@uta.edu

** Course:** Differential Equations and Linear Algebra,
Math 3319, Section 003, Course number 22282

** Office Hours:** 4:30-5:20 pm, Tuesday and Thursday, and by appointment

** Classroom:** PKH 107

** Classtime:** 5:30-6:50 pm Tuesday and Thursday

**Textbook:** Stephen Goode and Scott Annin,
Differential Equations and Linear Algebra, 3rd ed., Pearson, 2007.

**Coverage: ** The following sections in the textbook:
**1.1:** How differential equations arise
**1.2:** Basic ideas and terminology
**1.4:** Separable differential equations
**1.6:** First-order linear differential equations
**1.8:** Change of variables
**1.9:** Exact differential equations
**2.1:** Matrices: definitions and notations
**2.2:** Matrix algebra
**2.3:** Terminology for systems of linear equations
**2.4:** Elementary row operations
**2.5:** Gaussian elimination
**2.6:** Inverse of a square matrix
**3.1:** Definition of the determinant
**3.2:** Properties of determinants
**3.3:** Cofactor expansions
**4.1:** Vectors in **R**^{n}
**4.2:** Definition of a vector space
**4.3:** Subspaces
**4.4:** Spanning sets
**4.5:** Linear dependence and linear independence
**4.6:** Bases and dimensions
**5.1:** Definition of a linear transformation
**5.6:** Eigenvalue/eigenvector problem
**5.9:** Matrix exponential function
**6.1:** General theory of linear differential equations
**6.2:** Constant-coefficient homogeneous linear differential equations
**6.3:** Method of undetermined coefficients
**7.8:** Matrix exponential and systems of differential equations

** Grading: **
Letter grades will be assigned based on the three exam grades, with the lowest exam score contributing
20% and the other two exam scores each contributing 40% to the course grade.
The grade out of 100 on the first exam is equal to 10 √ 10(X_{1} + 1)/3 , where X_{1} is the number of correct answers on the first exam.
The grade out of 100 on the second exam is equal to 10 √ 10 X_{2}/3 , where X_{2} is the number of correct answers on the second exam.
The grade out of 100 on the third exam is equal to 10 √ 10 X_{3}/3 , where X_{3} is the number of correct answers on the third exam. All
grades out of 100 will correspond to the following scale: 0 __<__
F< 60,
60 __<__ D < 70, 70 __<__ C < 80, 80 __<__ B
< 90, 90 __<__ A __<__ 100.

**Exams:** Exam 1: February 16; Exam 2: April 6; Exam 3: May 9.

**Prerequisites:** A grade of C or better in Math 2326 or concurrent enrollment.

**Practice problems:**
**1.1:** 1, 3, 5, 7, 9, 11
**1.2:** 1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 25, 27, 31, 35, 37, 39
**1.4:** 1, 3, 5, 7, 15
**1.6:** 1, 3, 5, 7, 9, 15, 17
**1.8:** 1, 3, 5, 7, 9, 11, 13, 15
**1.9:** 1, 3, 5, 7, 11, 13, 15
**2.1:** 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 27
**2.2:** 1, 3, 7, 9, 11, 13, 15, 17, 19, 27, 31, 33, 37, 39, 43
**2.3:** 1, 3, 7, 9, 11, 13, 15, 17
**2.4:** 1, 3, 5, 9, 11, 15, 19, 21, 25
**2.5:** 1, 3, 5, 7, 13, 17, 21, 33, 37
**2.6:** 1, 3, 7, 9, 11, 15
**3.1:** 9, 11, 13, 17, 19, 21
**3.2:** 1, 3, 9, 15, 19, 21, 23, 25, 27, 29, 37, 39
**3.3:** 1, 3, 5, 7, 9, 11, 15, 41
**4.1:** 1, 3
**4.2:** 1, 3, 5, 7, 9, 11
**4.3:** 3, 5, 7, 9, 11, 13, 17, 19, 21
**4.4:** 1, 3, 5, 7, 11, 13, 17, 21, 23, 25
**4.5:** 1, 3, 5, 7, 13, 15, 19, 21, 29, 31
**4.6:** 3, 5, 7, 13, 15, 17, 21, 23
**5.1:** 1, 3, 5, 7, 9, 11, 13
**5.6:** 1, 3, 9, 11, 13, 15, 17, 21, 23, 25
**5.9:** 1, 3, 5, 7, 9
**6.1:** 1, 3, 17, 21, 23, 27, 31, 33, 35, 37
**6.2:** 5, 7, 9, 13, 17, 19, 21, 23, 29, 31, 33
**6.3:** 17, 19, 31
**7.8:** 1, 3, 5, 7

**Information on Exam 1:**
The exam will contain 30 questions (the first 15 are true/false questions and the remaining are
multiple-choice questions with 4 options to choose from). No materials
are allowed during the exam besides a pencil or a pen; blank sheets are
provided; the answer sheet is provided and will be collected at the end of
the exam. The coverage
will contain Sections 1.1, 1.2, 1.4, 1.6, 1.8, and 1.9 in the
textbook, and the following topics are emphasized:
- ODE: general solution, particular solution, explicit solution, implicit solution; arbitrary constants, initial conditions
- First-order ODEs: linear, separable, exact, homogeneous, Bernoulli; methods to solve such ODEs
- First-order linear ODEs: standard form, an integrating factor
- Differential, exact differential, total differential; criteria for exactness
- Substitution for Bernoulli equations, substitution for homogeneous equations
- Linear ODEs: nonhomogeneous term, homogeneous linear ODE, superposition principle, general solution, particular solution
- Linear
*n*th-order ODEs: with constant coefficients, Cauchy-Euler equations; functions satisfying such homogeneous linear ODEs
- Applications: LRC circuits, mass-spring systems

Answer key for Exam 1

**Information on Exam 2:**
The exam will contain 30 questions (the first 15 are true/false questions and the remaining are
multiple-choice questions with 4 options to choose from). No materials
are allowed during the exam besides a pencil or a pen; blank sheets are
provided; the answer sheet is provided and will be collected at the end of
the exam. The coverage
will contain Sections 2.1-2.6, 3.1-3.3, and 4.1-4.6 in the
textbook, and the following topics are emphasized:
- solving linear systems, Cramer's rule
- reduced row echelon form, elementary row operations
- matrices, rows and columns, matrix size
- matrix algebra (adding matrices, multiplying matrices, multiplying a matrix with a scalar)
- square matrix, identity matrix, diagonal matrix, upper and lower triangular matrices
- square of a matrix, cube of a matrix
- trace of a matrix, transpose of a matrix, inverse of a matrix, null space of a matrix
- permutation tensor
- determinant, properties of determinants
- vectors, vector space, subspace
- basis for a vector space, dimension of a vector space
- linear dependence/independence of a set of vectors
- linear combination of vectors
- vector space of
*m*×*n* matrices
- vector space of polynomials of degree
*n* or less

Answer key for Exam 2

**Information on Exam 3:**
The third exam, equivalent to the final exam, will be given during 5:30-8:00 pm on Tuesday,
May 9, 2017 in the standard classroom (PKH 107). The format of the exam will be
similar to the format of the first two exams.
The exam will contain 30 questions (the first 15 are true/false questions and the remaining are
multiple-choice questions with 4 options to choose from). No materials
are allowed during the exam besides a pencil or a pen; blank sheets are
provided; the answer sheet is provided and will be collected at the end of
the exam. The coverage
will contain Sections 5.6, 5.9, 6.1, 6.2, 6.3, 7.8 in the
textbook, but the knowledge of 2.1, 2.2, 2.4, 2.5, 2.6, 3.1, 3.2, 3.3, 4.5 is
also needed. The following topics are emphasized:
- eigenvalues and eigenvectors of a square matrix
- multiplicity of eigenvalues
- linearly independent eigenvectors, number of linearly independent eigenvectors
- complex and real eigenvalues
- determinant, trace, inverse of matrices in regard to eigenvalues
- real symmetric matrices in regard to eigenvalues
- diagonal, upper triangular, lower triangular matrices in regard to eigenvalues
- matrix exponentials and their properties
- linear ODEs and their general solutions
- linearly independent solutions to a homogeneous linear ODE
- operator notation
*D* for *d/dx*
- linear homogeneous ODEs with constant coefficients
- the method of undetermined coefficients
- solving linear homegeneous system of ODEs by using matrix exponentials

Answer key for Exam 3

*Tuncay Aktosun *
aktosun@uta.edu

Last modified: May 12, 2017