Math 316 Fall 2016

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Dec 11 2016:

Here is another proof that the real and imaginary parts of an eigenvector of a complex eigenvalue of a real matrix are linearly independent. And here is an example calculation involving a system whose coefficient matrix has a defective eigenvalue.

Assigned homework problems from Sections 9.6-9.7 can be found in the Homework Problems column of the class schedule. These are for your use in preparing for the final exam, but you don't have to hand them in.

This week: Liapunov's direct method for stability analysis. Periodic solutions and limit cycles. Final exam.

Special office hours this week: Tuesday 1-3 PM and Wednesday 3-4 PM (as usual) plus Thursday 3-5 PM.

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Home Page for Math 316 Fall 2016

Textbook:

Elementary Differential Equations, by William E. Boyce and Richard C. DiPrima, 10th edition, John Wiley and Sons, 2012.

You are expected to read the sections of the textbook listed in the class schedule thoroughly and carefully in advance of the indicated class. There will be quizzes based on the required reading.

Important: you may alternately get the textbook by the same authors entitled Elementary Differential Equations and Boundary Value Problems, as this just has two additional chapters at the end that while very interesting are not covered in Math 316. Whichever book you get, make sure to get the 10th edition.

Prerequisites:

Math 215, 255, or 285 (Calculus III) and Math 217 (Linear Algebra).

Class Meetings:

Mondays, Wednesdays, and Fridays 1:10 - 2 PM (section 1) or 2:10 - 3 PM (section 2) in 4088 East Hall. Occasional computer labs (see the class schedule) will be held during class time in room 2000 of the Shapiro Undergraduate Library.

Please come only to the section in which you are enrolled.

Office hours:

Tuesdays 1-3 PM and Wednesdays 3-4 PM in 5826 East Hall, or by appointment. You are encouraged to take advantage of this opportunity.

Grading and Course Policies:

Students will be evaluated on the basis of

Reading assignments/quizzes: 10%.
Homework and lab assignments: 10%.

Homework problems: the exam questions will be taken from versions of the posted homework problems. Therefore to succeed in the course it is necessary to understand and be able to solve the homework problems on your own, without the help of available solution manuals etc. You may exploit such tools to help check your understanding, knowing however that you will need to be able to do it all yourself on the exams. As indicated on the class schedule, you will have regular opportunities to choose problems that you are having the most trouble with and write up your partial solutions with your questions. You will then receive both homework credit and written feedback on your paper. You are also encouraged to take advantage of office hours to learn how to solve the problems.
Labs: lab classes will be held in room 2000 of the Shapiro Undergraduate Library (the glass-enclosed mac lab on the second floor). Source files you will need for the lab will be posted on the class schedule before the class meets. You are expected to read through the lab manual before class. A lab writeup is due in class following the lab as indicated on the class schedule.

Exams:
- Midterm Exam I: 10%. Wednesday, September 21, 6-7 PM, in 1360 East Hall.
- Midterm Exam II: 30%. Wednesday, October 26, 6-7 PM, in 1360 East Hall.
- Final Exam: 40%. Combined exam (both sections): Friday, December 16, 1:30-3:30 PM in 1300 Chemistry.
Calculators, books, and notes are not allowed on any exams. Exam questions will be eerily similar to or even identical to posted homework problems.

Grades given on individual quizzes, homeworks, labs, and exams will not be "curved". However the historical average cumulative grade for Math 316 is about a "B", and you should expect a similar statistic for our class.

Our class is carefully structured so that students will see every topic at least three times before an exam: first in the required reading (on which pop quizzes will be based), second in lecture, and third in working posted homework problems after the lecture.

Active participation in class is an important key to success in Math 316. Attendence of all lectures, labs, and exams is expected. Make-ups will not be given except in truly extraordinary circumstances.

Statement on accomodation of disabilities: If you think you need an accommodation for a disability, please let me know as soon as possible. In particular, a Verified Individualized Services and Accommodations (VISA) form must be provided to me at least two weeks prior to the need for a test/quiz accommodation. The Services for Students with Disabilities (SSD) Office (G664 Haven Hall; http://ssd.umich.edu/) issues VISA forms.

Schedule (subject to modification)
Week	Date	In Class	(Quizzable) Reading Assignments. Read before class.	Homework Problems. Do after class.	Exams
Week 1	Wednesday, September 7	Lecture 1: Sections 1.1-1.3. What is a differential equation? Mathematical modeling, direction fields for first-order equations.	Sections 1.1-1.3 and 2.1.	Please fill out the student data form. Problems for Sections 1.1-1.3.
Week 1	~~Friday, September 9~~ Class cancelled to due power outage.	~~Lecture 2: Section 2.1.~~ ~~Integrating factors for first-order linear equations.~~	~~Sections 2.2-2.3 and 2.5.~~	~~Problems~~ ~~for Section 2.1.~~
Week 2	Monday, September 12	Lecture 2: Section 2.1. Integrating factors for first-order linear equations.	Sections 2.2-2.3 and 2.5.	Problems for Section 2.1.
	Tuesday, September 13 Special date, time, and location: 6-7 PM in 1372 East Hall.	Lecture 3: Sections 2.2-2.3, 2.5. Separable and autonomous first-order equations and applications.	Sections 2.4 and 2.8.	Problems for Sections 2.2-2.3 and 2.5.
	Wednesday, September 14	Lecture 4: Sections 2.4 and 2.8. Differences between linear and nonlinear first-order equations. Conditions for existence and uniqueness of solutions of initial-value problems. Picard iteration.	Sections 3.1 and 3.2.	Problems for Sections 2.4 and 2.8.
	Friday, September 16	Lab 1: In room 2000 of the Shapiro Undergraduate Library. Using Mathematica to study differential equations. Exploring existence and uniqueness criteria and Picard iterates. Files needed: Lab1.nb (lab notebook) and UMMathDiffEq.m. Homework problems through Section 2.8 due.	Lab1 notebook.
Week 3	Monday, September 19	Lecture 5: Sections 3.1-3.2. Second-order linear homogeneous equations with constant coefficients. Characteristic equations and superposition principle. Independence of solutions and the Wronskian. Lab 1 due.	Section 3.3.	Problems for Sections 3.1-3.2.
	Wednesday, September 21	Review for Midterm I.	Midterm I Review Sheet.		Midterm I. 6-7 PM in 1360 East Hall. Covers Chapters 1 and 2.
	Friday, September 23	Lecture 6: Section 3.3. Complex roots of the characteristic equation.	Section 3.4.	Problems for Section 3.3.
Week 4	Monday, September 26 Drop Deadline	Lecture 7: Section 3.4. Repeated roots of the characteristic equation. The reduction of order method.	Section 3.5.	Problems for Section 3.4.
	Wednesday, September 28	Lecture 8: Section 3.5. Nonhomogeneous second-order linear equations. Solution structure (particular plus general homogeneous). Finding particular solutions by the method (applicable to certain special equations) of undetermined coefficients.	Section 3.6.	Problems for Section 3.5.
	Friday, September 30	Lab 2: In room 2000 of the Shapiro Undergraduate Library. Using Mathematica to study and compare linear and nonlinear differential equations. Files needed: Lab2.nb (lab notebook) and UMMathDiffEq.m.	Lab 2 notebook.
Week 5	Monday, October 3	Lecture 9: Section 3.6. The (general) method of variation of parameters for nonhomogeneous second-order linear equations. Lab 2 due.	Sections 3.7-3.8.	Problems for Section 3.6.
	Wednesday, October 5	Lecture 10: Sections 3.7-3.8. Mechanical and electrical vibrations. Damping and periodic forcing. Transient and steady-state response. Resonance. Video of resonant forced vibrations of the Tacoma Narrows Bridge and its eventual collapse in 1940. Corresponding Wikipedia entry.	Sections 4.1-4.4.	Problems for Sections 3.7-3.8.
	Friday, October 7	Lecture 11: Sections 4.1-4.4. Generalization to higher-order linear equations. Homework problems through Section 3.8 due.	Sections 5.1-5.2.	Problems for Sections 4.1-4.4.
Week 6	Monday, October 10	Lecture 12: Sections 5.1-5.2. Review of power series. Power series solutions of differential equations.	Section 5.3.	Problems for sections 5.1-5.2.
	Wednesday, October 12	Lecture 13: Section 5.3. Convergence of power series solutions about ordinary points.	Section 5.4.	Problems for Section 5.3.
	Friday, October 14	Lecture 14: Section 5.4. Euler-type equations and regular singular points. Homework problems through Section 4.4 due.	Section 5.5.	Problems for Section 5.4.
Week 7	Monday, October 17	Fall Break
	Wednesday, October 19	Lecture 15: Section 5.5. Series expansions of solutions near regular singular points. The method of Frobenius.	Sections 6.1-6.2.	Problems for Section 5.5.
	Friday, October 21	Lecture 16: Sections 6.1-6.2. Laplace transforms. Definition and use in studying linear initial-value problems. Homework problems through section 5.5 due.	Sections 6.3-6.4.	Problems for Sections 6.1-6.2.
Week 8	~~Monday, October 24~~ Class postponed (to be made up next week).	Lecture 17: Sections 6.3-6.4. Laplace transforms of step functions and applications to differential equations with discontinuous forcing.	~~Section 6.5.~~	~~Problems for Sections 6.3-6.4.~~
	Wednesday, October 26	Review for Midterm II.	Midterm II Review Sheet.		Midterm II. 6-7 PM in 1360 East Hall. Covers Chapters 1-5.
	Friday, October 28	Lecture 17: Sections 6.3-6.4. Laplace transforms of step functions and applications to differential equations with discontinuous forcing.	Section 6.5.	Problems for Sections 6.3-6.4.
Week 9	Monday, October 31	Lecture 18: Section 6.5. Impulsive forcing. Delta functions and their Laplace transforms.	Section 6.6.	Problems for Section 6.5.
	Tuesday, November 1 Special date, time, and location: 6-7 PM in 1068 East Hall.	Lecture 19: Section 6.6. Convolution of two functions and the connection with Laplace transforms.	Sections 7.1-7.3.	Problems for Section 6.6.
	Wednesday, November 2	Lab 3: In room 2000 of the Shapiro Undergraduate Library. Using Mathematica and Laplace transforms to study damped oscillators. File needed: Lab3.nb (lab notebook). Homework problems through Section 6.6 due.	Lab 3 notebook.
	Friday, November 4	Lecture 20: Sections 7.1-7.3. Introduction to first-order systems of linear differential equations. Connection with linear algebra. Lab 3 due.	Section 7.4.	Problems for Sections 7.1-7.3.
Week 10	Monday, November 7	Lecture 21: Section 7.4. Homogeneous systems. Superposition principle. Fundamental sets of solutions. Wronskian determinant.	Section 7.5.	Problems for Section 7.4.
	Wednesday, November 9	Lecture 22: Section 7.5. Constant coefficient linear systems. Eigenvalues/eigenvectors and the phase plane.	Section 7.6.	Problems for Section 7.5.
	Friday, November 11	Lecture 23: Section 7.6. Complex eigenvalues.	Section 7.7.	Problems for Section 7.6.
Week 11	Monday, November 14	Lecture 24: Section 7.7. Fundamental solution matrices and matrix exponentials for constant-coefficient systems.	Section 7.8.	Problems for Section 7.7.
	Wednesday, November 16	Lecture 25: Section 7.8. Repeated eigenvalues and generalized eigenvectors.	Section 7.9.	Problems for Section 7.8.
	Friday, November 18	Lecture 26: Section 7.9. Nonhomogeneous first-order linear systems.	Sections 8.1-8.2.	Problems for Section 7.9.
Week 12	Monday, November 21	Lecture 27: Sections 8.1-8.2. Numerical (computer) methods for approximate solution of differential equations. Euler's tangent line method and improvements. Homework problems through Section 7.9 due.	Section 8.3.	Problems for Sections 8.1-8.2.
	Wednesday, November 23	Lecture 28: Sections 8.2-8.3. Improvements to Euler's method. The Runge-Kutta method.	Section 9.1.	Problems for Section 8.3.
	Friday, November 25	Thanksgiving Holiday
Week 13	Monday, November 28	Lab 4: In room 2000 of the Shapiro Undergraduate Library. Using Mathematica to implement numerical methods for differential equations and to study their accuracy. Files needed: Lab4.nb (lab notebook) and UMMathDiffEq.m.	Lab 4 notebook.
	Wednesday, November 30	Lecture 29: Section 9.1. Review of linear first-order systems in the phase plane. Lab 4 due. Homework problems through Section 8.3 due.	Section 9.2.	Problems for Section 9.1.
	Friday, December 2	Lecture 30: Section 9.2. Autonomous nonlinear systems. Equilibria and stability.	Section 9.3.	Problems for Section 9.2.
Week 14	Monday, December 5	Lecture 31: Section 9.3. Effect of perturbations from equilibrium. Locally linear systems and stability analysis.	Sections 9.4-9.5.	Problems for Section 9.3.
	Wednesday, December 7	Lecture 32: Sections 9.4-9.5. Applications: competing species and predator/prey models.	Sections 9.6-9.7.	Problems for Sections 9.4-9.5.
	Friday, December 9	Lab 5: In room 2000 of the Shapiro Undergraduate Library. Using Mathematica to study nonlinear autonomous systems. Applications to economics. Files needed: Lab5.nb (lab notebook) and UMMathDiffEq.m. Homework problems through Section 9.5 due.	Lab 5 notebook.
Week 15	Monday, December 12	Lecture 33: Sections 9.6-9.7. Liapunov's direct method for stability analysis. Periodic solutions and limit cycles of nonlinear systems. Lab 5 due.		Problems for Sections 9.6-9.7.
Week 15	Friday, December 16		Final Exam Review Sheet.		FINAL EXAM

Peter D. Miller

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