math254

Math 254-DIFFERENTIAL EQUATIONS

Weekly Schedule:
Mondays: 10:40-12:30 M-13
Wednesdays: 08:40-10:30 M-13
Office Hours:
TBA

Prerequisites: Math 154, Math 261

Course Objectives: The objectives of this course are to introduce the student with the concept of a differential equation, basic techniques for solving certain classes of differential equations, especially those which are linear, and making connections between the qualitative features of the equation and the solutions. Connections to problems from the physical world are emphasized. As well as ordinary differential equations, the course aims to introduce the students to certain partial differential equations.

Course Learning Outcomes: By the end of this course, a student will:

classify and identify different types of differential equations,
explicitly solve several important classes of ordinary differential equations and interpret their qualitative behavior,
apply ideas from linear algebra in order to solve single linear ordinary differential equations and systems of such equations,
model certain physical phenomena using differential equations and reinterpret their solutions physically,
use power series methods to solve second order linear differential equations
apply the Laplace transform for solving differential equations,
use the method of separation of variables in order to solve some basic partial differential equations via Fourier series

Instructional Methods: Instruction will be in class lectures delivered by the instructor.

Course Textbook(s): “Elementary Differential Equations and Boundary Value Problems”, Boyce, W. E., DiPrima, R. C., 9th ed.

Tentative Weekly Outline:
Section and page numbers below are from the textbook, Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima, 9th ed., 2010.

Weeks	Topics	Suggested Problems
1	Chapter 1. §1.1, §1.3: Introduction, Direction Fields Chapter 2. First Order Differential Equations §2.2: Separable equations (also homogeneous equations - see49#30). §2.1: Linear equations; Method of integrating factors.	§1.1: 6, 8, 15-20 §1.3: 1, 2, 5, 9, 10, 16, 24, 28 §2.2: 3, 5, 7, 11, 12, 23, 31, 34, 37 §2.1: 2, 3, 6, 10, 11, 15,
2	§2.3: Modeling with first order equations §2.4: Differences between linear and nonlinear equations	§2.3: 1, 2, 3, 4, 5, 13, 16, 18, 19 §2.4: 2, 4, 6, 7, 8, 10, 13, 14, 15, 22, 23, 24, 25, 26
3	§2.6: Exact equations and integrating factors. §2.8: Existence and Uniqueness Theorem	§2.6: 1, 2, 4, 7, 12, 14, 16, 19, 22, 23, 24, 31, 32 §2.8: 1, 7, 8, 13, 14, 15, 16, 17, 18, 19
4	Chapter 7. Systems of First Order Linear Equations §7.3: Systems of linear algebraic equations; Linear independence, eigenvalues, eigenvectors. §7.4: Basic theory of systems of first order linear equations.	§7.3: 2, 3, 5, 9, 10, 11, 13, 14, 15, 19, 21, 22, 23, 24, 25 §7.4: 1, 5, 8, 9
5	§7.5: Homogeneous linear systems with constant coefficients §7.6: Complex eigenvalues.	§7.5: 11, 12, 13, 15, 17, 30, 32 §7.6: 1a, 2a, 4a, 6a, 8, 10, 25, 26
6	§7.7: Fundamental matrices. §7.8: Repeated eigenvalues. §7.9: Nonhomogeneous linear systems (variation of parameters only)	§7.7: 5, 7, 9, 10 §7.8: 1c, 4c, 5, 6, 8a, 12a, 16, 19, 20abc, 21abc §7.9: 1, 2, 3, 5, 10, 11, 12, 13, 16
7	Chapter 4. Higher Order Linear Equations §4.1: General theory of nth order linear equations §4.2: Homogeneous equations with constant coefficients.	§4.1: 7, 9, 10, 14, 16, 18, 25 §4.2: 12, 13, 18, 21, 22, 30, 33, 35, 40
8	§4.3: The method of undetermined coefficients. §4.4: The method of variation of parameters	§4.3: 1, 3, 4, 7, 8, 10, 11, 12, 21 §4.4: 1, 2, 4, 7, 9, 10, 14, 16
9	§3.7: Mechanical and electrical vibrations. §3.8: Forced Vibrations Chapter 5. Series Solutions of Second Order Linear Equations §5.1: Review of Power Series §5.2: Series Solutions Near an Ordinary Point, Part I §5.3: Series Solutions Near an Ordinary Point, Part II	§3.7: 2, 6, 8, 10, 19, 24, 27, 29, 30 §3.8: 2, 3, 5, 9, 11, 16, 18, 19, 20 §5.1: 3, 4, 5, 12, 13, 15, 22, 24, 27, 28 §5.2: 2, 3, 5, 7, 10, 14, 15, 21
10	§5.4: Euler Equations, Regular Singular Points §5.5: Series Solutions Near a Regular Singular Point, Part I	§5.4: 17, 20, 22, 29, 31, 36 §5.5: 3, 6, 8, 11, 12, 13, 14
11	Chapter 6. The Laplace Transform §6.1: Definition of the Laplace transform §6.2: Solution of initial value problems. §6.3: Step functions	§6.1: 3, 4, 7, 9, 17, 20, 21, 24, 26, 27 §6.2: 1, 2, 5, 6, 8, 11, 14, 17, 19, 23, 35, 37 §6.3: 1, 2, 4, 5, 7, 9, 10, 12, 13, 14, 20, 22, 24, 33, 37
12	§6.4: Differential equations with discontinuous forcing functions. §6.5: Impulse functions. §6.6: The convolution integral	§6.4: 2, 4, 5, 6, 9, 10, 12, 13, 16, 18 §6.5: 2, 5, 6, 7, 11, 12, 15, 17, 19 §6.6: 3, 7, 8, 10, 11, 13, 15, 19, 21, 22, 23, 25
13	Chapter 10. Partial Differential Equations and Fourier Series §10.1: Two-point boundary value problems. §10.2: Fourier series.	§10.1: 1, 2, 6, 8, 11, 14, 15, 17, 20, 22 §10.2: 1, 2, 3, 8, 10, 14, 16, 18, 23, 28
14	§10.3: The Fourier convergence theorem §10.4: Even and odd functions. §10.5: Separation of variables, heat conduction in a rod	§10.3: 1, 3, 5, 6 §10.4: 7, 8, 10, 11, 17, 19, 21, 23, 26 §10.5: 1, 2, 3, 5, 7, 8, 10, 12, 18, 19, 21, 22

Grading: There will be two midterm exams and a final exam,

Midterm Exam 1: 30% April 14, 2022, Thursday, at 17:40
Midterm Exam 2: 30% May 24, 2022, Tuesday, at 17:40
Final Exam: 40%

Course Policies:
Class Attendance: Attendance of lectures is required.
Make-up Policy: In order to be eligible to enter a make-up examination for a missed examination, a student should have a documented or verifiable, and officially acceptable excuse. A student cannot get make-up examinations for two missed exams. The make-up examination for all exams will be after the final exam, and will include all topics.
Final Exam Entrance Conditions NA Policy: If you miss all midterm exams and final exam, you will receive a grade of NA for the course.