Math 0290: Differential Equations

Instructor: Catalin Trenchea
Lectures: MWF 1:00-1:50PM, G30 Benedum Hall

Office Hours: Monday and Wednesday 2:00pm-3:15pm, and by appointment (also via zoom)
Office: 612 Thackeray Hall
E-mail: trenchea@pitt.edu

Overview

Differential equations represent an important branch of mathematics. Many of their properties have been understood mathematically and they have a history of being successfully applied to important problems in all areas of science and engineering. This course will introduce primarily linear, first-order, and second-order differential equations. Solution techniques for separable equations and homogeneous and inhomogeneous equations as well as a range of modeling-based applications arising in the context of engineering, physics and chemistry will be presented. The application of Laplace transforms to differential equations, systems of linear differential equations, linearization of nonlinear systems, and phase plane methods will be covered. Fourier series, a useful tool in signal processing, will also be introduced, and we will discuss how the Fourier series arises in solving the famous heat equation by separation of variables. The idea of approximating and visualizing solutions using a computer, such as with Matlab, will be introduced early in the term and students are expected to use Matlab as a resource in their work for this course.

Textbooks

There is a link in Canvas which includes the purchase of the electronic version of the textbook onto your tuition statement if you do not `opt out'. This purchase offers more than what is necessary. The only requirement to this course is the textbook. Students may choose to use the first edition of the text or a used second edition, which may be available at a lower cost. If you wish to do that, you should choose the `opt out' option prior to the add/drop deadline and visit http://calculus.math.pitt.edu and click the Textbook information link.

Tutoring: The Mathematics Department offers a free tutoring service. The Math Assistance Center (MAC) is located on the second floor of the O’Hara Student Center. Tutoring services and tutoring hours will be posted outside the MAC as well as on the web at MAC.

Grades

Grading:
Assignments: Weekly homework assignments will be collect at the beginning of the lecture every Monday. The assignment grade will be 20% of the course grade.

Midterm Exams: There will be two in class midterm examinations given. The second midterm will not be cumulative to the first. In other words, the second midterm will only cover course material not covered by first midterm exam. Each midterm exam grade will be 20%(x2) of the course grade.

Final Exam: The final exam grade will be 40% of the course grade and will take place during exams week. Your course grade will not exceed your final exam grade by more than one letter grade.

A/A-:90-100%, B/B±: 80-89%, C/C±: 70-79%, D/D+: 60-69%, F: < 60%

Some sections may deviate slightly from this recipe. Any deviations will be announced by your instructor at the beginning of the term.

MATLAB : Computers are often used to study solutions to differential equations in physics, biology, chemistry, and engineering. Right from the outset, we will discuss how Matlab can help us to visualize the behavior of solutions of differential equations and to approximate these solutions and we will give an introduction to numerical solution techniques. Matlab will not be available on quizzes/exams, however, and will not factor heavily into statements of homework problems; mostly, it is a tool that can help you understand the material better and check your solutions.

Homework policies

Students are required to complete the homework problems; very few students can learn this material without constant practice. Students are welcome to work together on homework. However, each student must turn in his or her own assignments, and no copying from another student's work is permitted. Deadline extensions for homework will not be given. Students are encouraged to discuss with your professor about homework problems if you'd like additional feedback.

Final Exam Policy

All day sections will take a departmental Final Exam on

12/11/2024, Wednesday 4:00PM - 5:50PM, 104 Thaw Hall.

All students must take the final exam on the day and time scheduled by the registrar.
Evening sections will meet through final exam week, and the final exam will be given during the last one or two scheduled class periods.

If any assessments will be administered online, proctoring might be done via ZOOM and a video connection will be required.

Final Grade Policy

Your final grade should not exceed your final exam grade by more than one letter grade.

Office Hours

Your instructor will announce his office hours.

Disability Resource Services

If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services (DRS), 140 William Pitt Union, 412-648-7890, drsrecep@pitt.edu, (412) 228-5347 for P3 ASL users, as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.

Academic Integrity

The University of Pittsburgh Academic Integrity Code is available at https://www.provost.pitt.edu/faculty/academic-integrity-freedom/academic-integrity-guidelines. The code states that "A student has an obligation to exhibit honesty and to respect the ethical standards of the academy in carrying out his or her academic assignments." The website lists examples of actions that violate this code. Students are expected to adhere to the Academic Integrity Code, and violations of the code will be dealt with seriously.

On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.

This is especially notable during this period. Cheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction.
Please note, in particular, that Pitt has a data sharing arrangement with Chegg.com that enables us to identify in- stances in which Chegg.com has been used to cheat on assessments. Consequences of being caught in this academic integrity violation have included zero scores on assessments and F grades for the course.

Diversity and Inclusion

The University of Pittsburgh does not tolerate any form of discrimination, harassment, or retaliation based on disability, race, color, religion, national origin, ancestry, genetic information, marital status, familial status, sex, age, sexual orientation, veteran status or gender identity or other factors as stated in the University’s Title IX policy. The University is committed to taking prompt action to end a hostile environment that interferes with the University’s mission. For more information about policies, procedures, and practices, see: https://www.diversity.pitt.edu/civil-rights-title-ix-compliance/policies-procedures-and-practices.

Classroom Recording

To ensure the free and open discussion of ideas, students may not record classroom lectures, discussion and/or activities not already recorded by the instructor, without the advance written permission of the instructor, and any such recording properly approved in advance can be used solely for the student’s own private use.

Lectures could be recorded by the instructor, and this may include student participation. Students are not required to participate in the recorded conversation. The recorded lecture may be used by the faculty member and the registered students only for internal class purposes and only during the term in which the course is being offered. Recorded lectures will be uploaded and shared with students through Canvas.

Copyright

Some of the materials in this course may be protected by copyright. United States copyright law, 17 USC section 101, et seq., in addition to University policy and procedures, prohibit unauthorized duplication or retransmission of course materials. See the Library of Congress Copyright Office and the University Copyright Policy.

Schedule and practice problems

Approximate schedule for lectures. References of the form a.b refer to sections in the main textbook.
(For midterms and final exams from previous years, please look at Eugene Trofimov's webpage.)

Week 1
August 26: Introduction to Differential Equations (DE)
dfield.jar
 1.1 Number 1-11. Homework: 1,2,5,7,11
August 28: First Order Initial Value Problems
2.1 Number 1-6, 12-15. Homework: 1,3,5,12,13,15
August 30: Numerical methods and computer tools including Matlab for DEs
6.1 Number 1-5 Homework: 3,5
Solutions

Week 2
September 4: Numerical Methods. Runge-Kutta Methods
6.2 Number 1-9. Homework: 5, 23
September 6: Numerical Methods. Numerical Error.
6.3 Number 1-6, 11-13.
Solutions

Week 3
September 9 Separable Equations
2.2 Number 1-22, 23-29, 33-35 Homework: 3,5,9,33
September 11: Models of Motion
2.3 Number 1-10 Homework: 9
September 13: First Order Linear Equations
2.4 Number 1-21 Homework: 5,15,19
Solutions

Week 4
September 16: Mixing Problems
2.5 Number 1-7, 9-10 Homework: 5, 9b
September 18: Electrical Circuits
3.4 Number 1-19 Homework: 1,3,5,7,11
September 20: Second Order Equations
4.1 Number 1-20, 26-30 Homework: 1,3,9,17
Solutions

Week 5
September 23: Linear Homogeneous Equations with Constant Coefficients
4.3 Number 1-36 Homework: 1,9,17,35
September 25: Harmonic Motion
4.4 Number 1-12, 14-16, 18 Homework: 1,7
September 27: Inhomogeneous second order equations. Undetermined Coefficients
4.5 Number 1-29 Homework: 1,5,11
Solutions

Week 6
September 30: Inhomogeneous second order equations. Undetermined Coefficients (continued)
4.5 (cont.) Number 1-29 Homework: 15,19
October 2: Inhomogeneous Equations. Variation of Parameters
4.6 Number 1-10 Homework: 1,3,5
October 4: Forced harmonic motion
4.7 Number 3-11 Homework: 3,11
Solutions

Week 7
October 7: Review
October 9: Midterm 1
October 11: Laplace Transform
5.1 Number 1-29 Homework: 7,13,15,29
Solutions

Week 8
October 14: Fall Break for students (No Classes)
October 16: Laplace Transform. Basic properties
5.2 Number 1-41 Homework: 5,11,19,29
October 18: The Inverse Laplace Transform
5.3 Number 1-36 Homework: 3,7,11,19
Solutions

Week 9
October 21: Using the Laplace Transform to solve DEs
5.4 Number 1-26 Homework: 7,11,21
October 23: Discontinuous Forcing Term
5.5 Number 1-25 Homework: 1,3,11,17
October 25: The Dirac Delta Function
5.6 Number 1-9 Homework: 2,3,5,7
Solutions

Week 10
October 28: Convolutions
5.7 Number 4-24 Homework: 6,8,10
October 30: Introduction to Systems
8.1 Number 1-16 Homework: 5,7,13,15
November 1: Systems (continued)
8.2 Number 1-6, 13-16 Homework: 11,13,15 (use pplane.jar)
Solutions

Week 11
November 4: Systems of differential equations, Constant coefficient homogeneous 2x2 systems
8.3 Number 1-6 Homework: 1,3,5
November 6: Linear Systems with Constant Coefficients
9.1 Number 1-8, 16-23 Homework: 3,5,17,19
November 8: Planar Systems
9.2 Number 1-27, 58-61 Homework: 3,13,15,59
Solutions

Week 12
November 11: Phase Plane Portraits
9.3 Number 20-23 Homework: 21
November 13: Nonlinear Systems: Equilibria, Linearization
10.1 Number 1-16 Homework: 3,7,15
November 15: Review
Solutions

Week 13
November 18: Midterm 2
November 20: Fourier series
12.1 Number 1-22 Homework: 5,7,13,17
November 22: Fourier Cosine and Sine Series
12.3 Number 1-32 Homework: 3,7,19,31
Solutions

Week 14
December 2: Heat Equation
13.1 Number 1-9
December 4: Separation of variables for the heat equation
13.2 Number 1-18
December 6: Separation of variables for the heat equation (continued)
13.2 Number 1-18
Solutions

December 9: Review