## Math 251H, Section 1: Ordinary and Partial Differential Equations (Spring 2018)

Instructor: Dr. Jessica M. Conway.
Lectures: MWRF 12:20 - 1:10pm, Huck Life Sciences Bldg 011.
Office hours: McAllister 332, Mondays 9-10:30am, Wednesdays 3-4:30, and by appointment.
Email: jmconway (at) psu (dot) edu
Phone: (814)863-9125

PSU offers Mathematics drop-in and online tutoring, ODEs/PDEs included!
Schedule and locations available at https://pennstatelearning.psu.edu/tutoring/mathematics.
Course Syllabus available HERE.

Text: Nagle, Saff, and Snider, Fundamentals of Differential Equations, 8th Edition.
We will cover chapters 1-2, 4-7, 9-10, and 12.2-3.

• We'll use software to understand direction fields and phase planes, available at http://math.rice.edu/~dfield/dfpp.html.
• MATLAB may be used for a few demonstration; if so, code will be provided. It is available for free as a WebApp at https://webapps.psu.edu/.
• Web-based software will be used for demonstrations in sections 4.9/4.10. It will be provided when the time comes.
• Dr. Wen Shen's Lecture Notes for Math 251/251H HERE. She is generously sharing them with us as an additional resource. There are LOTS of examples!
• Forced Vibrations web application: link.

• ### ANNOUNCEMENTS:

• (01/08/2018) Welcome to Math 251H!

• ### Exam Dates:

• Midterm test 1: Thursday, February 22, 2018; 6-7:15pm, 262 Willard.
• Midterm test 2: Monday, April 2April 4, 2018; 6-7:15pm, 220 Willard.
• Final exam: Thursday May 3, 2018; 8-9:50am, 060 Willard.

50 points (10%) from homework + 70 points (14%) from quizzes + 30 points (6%) from mini projects + 100 points (20%) from Midterm 1 + 100 points (20%) from Midterm 2 + 150 points (30%) from the Final Exam = 500 points (100%).

### Homework

• Due each week at the beginning of the Monday's class unless otherwise stated. For now, homework will only be checked for completion, but not graded.
• There will be 12-13 homework sets. Each homework will be worth 5 points, up to a total of 50 towards your final grade.
• Warning: if you all start "phoning it in", homework will be partially graded.

Late home work will NOT be accepted.

### Quizzes

• 15-20 minutes, every Wednesday, based on the previous week's homework.
• We'll drop the lowest two quizzes to calculate the final quiz grade.

• Quiz SOLUTIONS:
Quiz 1 solutions
Quiz 2 solutions
Quiz 3 solutions
Quiz 4 solutions
Quiz 5 solutions
Take home Quiz 6 and solutions.
Quiz 7 solutions
Quiz 8 solutions
Quiz 9 solutions
Quiz 10 solutions
Quiz 11 solutions
Quiz 12 solutions
Quiz 13 solutions
Quiz 14 solutions
Not for credit, Quiz 15 and solutions

### Exams

Old exams are available in a repository maintained by Dr. Zachary Tseng, here.

Midterm 1 prep
• Covers Sections 1.1-1.3, 2.1-2.6, 3.1-3.5, 4.1-4.7, 4.9-4.10. Each chapter has a nice and succinct summary, take advantage!
• 2016: midterm and solutions.
• Fall 2017: midterm and solutions.

• Midterm 2 prep
• Covers sections 6.1-6.2, 7.1-7.8, 5.1, 5.4, 9.3-9.8 (9.8: generalized eigenvectors only), 12.2.
• 2016: midterm and solutions.
• Fall 2017: midterm and solutions.
• REVIEW Monday April 2, 6-7:15pm, 262 Willard.

• Final exam prep
• Covers Sections 1.1-1.3, 2.1-2.6, 3.1-3.5, 4.1-4.7, 4.9-4.10, 6.1-6.2, 7.1-7.8, 5.1, 5.4, 9.3-9.8 (9.8: generalized eigenvectors only), 12.1-12.3, 10.1-10.7.
• 2016: exam and solutions.
• 2017: exam.

• ### Mini projects

Throughout the semester you will complete 3 mini-projects related to course material. Each will be worth 2% of your final mark.
• You may work in groups of 1-2.
• Topics will be listed below.

• Guidelines for project presentation: link.
Due dates: In class, Feb 16, Mar 30, Apr 27.

Mini-project 1, due in class Friday Feb. 16. Options:
Chapter 1, Project C.
Chapter 2, Projects A, B, G.
Chapter 3, Projects A, B, C, E, F, G, I**.
** requires sections 1.4/3.6 on numerics which we did not discuss in class, but you're welcome to work on anyway! I'm happy to help.
Mini-project 2, due in class Friday Mar. 30 Options:
Chapter 5, Projects A**, B, D, E, F**.
Chapter 6, Projects C**, D.
Chapter 7, Project B.
Chapter 9, Project B .
** requires some numerics which we did not discuss in class, but you're welcome to work on anyway! I'm happy to help.
Mini-project 3, due in class Friday Apr. 27 Options:
Chapter 10, Project A, B, C, D**.
Chapter 12, Project D, E, F.
Zombie project description and associated paper.
** requires numerical approaches which we did not discuss in class, but you're welcome to work on anyway! I'm happy to help.

### Documents

(1) Greek alphabet.
(2) Problem 2.5.5 solution and more.
(3) Population dynamics example, problem 3.2.12, 14, 15 solution and more.
(4) Three variation of parameters examples.
(5) Additional reduction of order example.
(6) Videos showing the applications of resonance: bending & breaking a glass, bending & breaking a bridge.
(7) Forced & free vibrations example.
(8) Refresher on polynomial long division: Khan Academy Video.
(9) Seventh order, linear, homogeneous ODE general solution example.
(10) For enrichment only, NOT course material: Handout on computing the inverse Laplace transform via complex integration. The handout shows the basic steps and discusses a couple of the theorems used; it also contains citations for proofs and further reading.
(11) Table of Laplace transforms: link. This table is the one that will be provided for quizzes and exams.
(12) Discontinuous forcing functions example.
(13) Laplace transforms of periodic functions without window functions: derivation.
(14) Same discontinuous forcing functions example using convolution.
(15) Examples of eigenvalue & eigenvector calculations: link.

### Lecture Schedule (tentative; will be updated as semester proceeds)

Jan 8 I. Introduction: what is a DE, order, linear and nonlinear, solutions, direction fields.
II. First Order Differential Equations: separable equations, integrating factors.
1.1-1.3, 2.1-2.3. 1.1: 7, 15
1.2: 7, 21, 27, 31
1.3: 1, 7, 10cd, 17
2.2: 3, 5, 15, 17, 25, 29, 33, 39
2.3: 3, 5, 7, 15, 19, 21, 23
Enrichment, not for credit: 1.2.19, 2.3.29
Lecture 1
Lecture 2
Lecture 3
Lecture 4
Jan 15 II. First Order Differential Equations: exact equations, special integrating factors, substitutions and transformations 2.4 - 2.6. 2.4: 1, 3, 5, 13, 17, 23, 25, 27
2.5: 1, 3, 5, 9, 11, 13, 20
2.6: 11, 15
Enrichment, not for credit: 2.4.32, 2.6.47
Lecture 5
Lecture 6
Lecture 7
Jan 22 II. First Order Differential Equations: substitutions and transformations
III. Modeling with First-order equations
2.6, 3.1-3.5 2.6: 1, 3, 7, 17, 27, 31, 42
3.2: 7, 14, 15, 21, 25
3.3: 5, 12
3.5: 1, 5
Enrichment, not for credit: 2.6.47, 3.3.15, 3.4.20
Lecture 8 and examples.
Lecture 9
Lecture 10
Lecture 11
Jan 29 IV. Second order linear equations: homogeneous equations: general solution, auxiliary equations, real, complex, an repeated roots. Nonhomogeneous equations: method of undetermined coefficients. 4.1-4.4 3.4: 9, 19, 25
4.1: 2
4.2: 7, 11, 17, 19, 27, 29, 46
4.3: 17, 23, 25, 31 (do not solve), 32, 33

Enrichment, not for credit: 4.2.21, 23, 35ab, 37; 4.3.19, 27
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Feb 5 IV. Second order linear equations: Nonhomogeneous equations: method of undetermined coefficients, variation of parameters. Variable-coefficient equations. 4.4-4.8 4.4: 1, 3, 5, 7, 15, 21, 23, 27, 29, 31
4.5: 1, 9, 15, 29, 35, 43
4.6: 5, 15
4.7: 1, 5,37 Enrichment, not for credit: 4.4.35, 4.8.15 (good intuition building!)
Lecture 16
Lecture 17
Feb 12 IV. Second order linear equations: forced vibrations and free vibrations.
4.9-4.10 4.7: 11, 13, 15, 45, 47
4.8: 11 (self-study)
4.9: 3, 5, 7, 11
4.10: 1, 3, 5, 9, 11
Enrichment, not for credit: 4.8.6 (conservation of energy!)

Lecture 19
Lecture 20
Lecture 21
Lecture 22
Feb 19 V. Higher order linear equations: Solving constant-coefficient higher-order linear equations.
VII. Laplace transforms: Definitions, properties.

MIDTERM I THURSDAY
6.1-6.2, 7.1-7.3 6.1: 5, 7, 8, 21
6.2: 1, 5, 9, 13, 19, 31
7.2: 3, 11, 13, 19, 23, 25, 27, 29

Lecture 25
Feb 26 VII. Laplace transforms: inverse, initial value problems, discontinuous functions. 7.3-7.6 7.3: 1, 7, 17, 21, 24, 31
7.4: 11, 19, 29, 35
7.5: 11, 13, 21, 27, 31, 37
7.6: 3, 5, 7 (step fns only)
Enrichment, not for credit: 7.3.22, 7.3.37, 7.4.37, 7.4.42, 7.4.43
Lecture 26
Lecture 27
Lecture 28
Lecture 29
Mar 5 SPRING BREAK
Mar 12 VII. Laplace transforms: convolution, impulses. 7.6-7.8 7.6: 11, 17, 19, 21, 27, 33, 39, 59
7.7: 3, 9, 13, 19, 21, 25, 27, 29, 31 (solve IVP only for 25, 27)
7.8: 3, 5, 9, 11, 17, 19, 23, 29
Enrichment, not for credit: 7.8.35
Lecture 30
Lecture 31
Lecture 32
Lecture 33
Mar 19 VI. Systems of differential equations: phase plane analysis and applications; solving homogeneous linear systems, linear algebra review. 5.1, 5.4, 9.3-9.5
5.4: 11, 18, 19, 27, 28, 33
9.3: 5, 9, 21, 27, 35
9.4: 1, 9, 15, 25
9.5: 3, 11, 17, 19, 31, 45

Lecture 34
Lecture 35
Lecture 36
Lecture 37
Mar 26 VI. Systems of differential equations: Solving linear systems, analysis of nonlinear systems. 9.5-9.8; 12.2. 9.6: 1, 13ab, 19
9.7: 9.7.1, 7, 25, 26a
12.2: 5, 13, 14, 15, 16, 19, 20 (13-20: find general solution too, and try to sketch by hand), 21
Enrichment, not for credit: 9.6.21, 9.8.17
Lecture 38
Lecture 39
Apr 2 VII. Nonlinear systems of ODEs.
MIDTERM II WEDNESDAY
12.3, 10.1 - 10.3 12.3: 11, 13, 15, 17, 19, 21 (also check nullclines for 13, 15) Lecture 42
Lecture 43
Lecture 44
Comp model phase plane
Apr 9 VIII. PDEs and Fourier Series: Separation of variables, Fourier series, Fourier sine and cosine series
10.1-10.4 10.2: 5, 13, 17, 21, 24., 27, 28., 30., 31, 33
10.3: 1, 3, 5, 7, 11, 15, 19, 23, 25a, 28, 37

Enrichment, not for credit: 10.3.29, 10.3.31-39
To plot partial sums and errors: Mathematica file.
Lecture 45
Lecture 46
Lecture 47
Lecture 48
Matlab Fourier series demo
Apr 16 VIII. PDEs and Fourier Series: Fourier sine and cosine series, Heat equation, Wave equation. 10.4-10.6 10.4: 3, 9, 15, 17
10.5: 9, 11, 13, 15, 19
10.6: 5, 7, 9, 11, 19
Enrichment, not for credit: 10.6.20-21
Lecture 49
Lecture 50
Lecture 51
Lecture 52
Matlab heat eq. demo
Matlab wave eq. demo
Apr 23 VIII. PDEs and Fourier Series: Laplace equation.

FINAL EXAM REVIEW
10.7 10.6: 15, 17
10.7: 3, 9, 11, 17, 23
Lecture 53
Lecture 54
Lecture 55

Jessica M. Conway / Department of Mathematics / Pennsylvania State University