### Math 0290: Differential Equations

#### 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

• Polking, Boggess and Arnold, Differential Equations with Boundary Value Problems, second edition, Pearson Prentice-Hall.

Your course grade will be determined as follows: Two midterm exams 40% (20% each), Final exam 40%, Homework 20%.
Assignments: Ten assignments will be given throughout the term. The best eight assignments will be used to compute the final assignment grade. 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.

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 component: The study of differential equations often uses computer algorithms to gain solutions to relevant problems in physics, biology, chemistry, and engineering. Several assignment problems will taken from the problem sets in the MATLAB supplemental textbook. These problems will be of use to the student in both acquiring a visual sense of differential equations and their solutions, as well as give an introduction into standard-practice techniques currently used in many disciplines.

#### 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 at a time and place scheduled by the registrar: 12/12/2018, Wednesday at 2:00PM - 3:50PM, in G29 Benedum Hall. Evening sections will meet through final exam week, and the final exam will be given during the last one or two scheduled class periods.

#### Office Hours

Your instructor will announce his office hours.

The University of Pittsburgh Academic Integrity Code is available at https://provost.pitt.edu/faculty-resources/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.

#### 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, 140 William Pitt Union, 412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.

#### 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:
Introduction to Differential Equations (DE),
numerical methods and computer tools
including Matlab for DEs
1.1 Number 1-11. Homework: 1,2,5,7,11
2.1 Number 1-6, 12-15. Homework: 1,3,5,12,13,15
6.1 Number 1-5 Homework: 3,5

Solutions

Week 2:
Numerics (cont.), Separation of variables
6.2 Number 1-9. Homework: 23
6.3 Number 1-6, 11-13.
2.2 Number 1-22, 23-29, 33-35 Homework: 3,5,9,33
Solutions

Week 3:
Modeling, linear first-order equations
2.3 Number 1-10 Homework: 9
2.4 Number 1-21 Homework: 5,15,19
2.5 Number 1-7, 9-10 Homework: 5, 9b
Solutions

Week 4:
Modeling (cont.), second order equations
3.4 Number 1-19 Homework: 1,3,5,7,11
4.1 Number 1-20, 26-30 Homework: 1,3,9,17
4.3 Number 1-36 Homework: 1,9
Solutions

Week 5:
Second order equations (cont.), harmonic motions
4.3 (cont) Number 1-36 Homework: 17,35
4.4 Number 1-12, 14-16, 18 Homework: 1,7
4.5 Number 1-29 Homework: 1,5,11
Solutions

Week 6:
Inhomogeneous second order equations
4.5 (cont.) Number 1-29 Homework: 15,19
4.6 Number 1-10 Homework: 1,3,5
4.7 Number 3-11 Homework: 3,11
Solutions

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

Week 8:
Laplace Transform (cont.)
5.2 Number 1-41 Homework: 5,11,19,29
5.3 Number 1-36 Homework: 3,7,11,19
5.4 Number 1-26 Homework: 7,11,21
Solutions

Week 9:
Laplace Transform (cont.)
5.5 Number 1-25 Homework: 1,3,11,17
5.6 Number 1-9 Homework: 2,3,5,7
5.7 Number 4-24 Homework: 6,8,10
Solutions

Week 10:
Systems of differential equations; pplane, dfield
8.1 Number 1-16 Homework: 5,7,13,15
8.2 Number 1-6, 13-16 Homework: 11,13,15 (use pplane.jar)
8.3 Number 1-6 Homework: 1,3,5
Solutions

Week 11:
Constant coefficient homogeneous 2x2 systems
9.1 Number 1-8, 16-23 Homework: 3,5,17,19
9.2 Number 1-27, 58-61 Homework: 3,13,15,59
9.3 Number 20-23 Homework: 21
9.4 Number 1-12
Solutions

Week 12:
Midterm 2, nonlinear systems
10.1 Number 1-16 Homework: 3,7,15
Solutions

Week 13-14:
Fourier series
12.1 Number 1-22 Homework: 5,7,13,17
12.3 Number 1-32 Homework: 3,7,19,31
12.4 Number 1-11 Homework: 3
Solutions

Week 15:
Separation of variables for the Heat equation
Review
13.2 Number 1-18