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Dec 10 2017:

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

Solutions for Midterm II are available here: Version 1 / Version 2. The average score was about 13/20.

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Home Page for Math 316 Fall 2017


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.


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

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; 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 6 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.

Friday, September 8

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 11

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 13 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 15

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 18

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 20 Review for Midterm I. Midterm I

Review Sheet.

Midterm I. 6-7 PM, 1360 East Hall. Covers Chapters 1 and 2.
Friday, September 22 Lecture 6: Section 3.3. Complex roots of the characteristic equation. Section 3.4. Problems for Section 3.3.
Week 4

Monday, September 25

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 27 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 29

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 2

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 4

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 6

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 9 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 11 Lecture 13: Section 5.3. Convergence of power series solutions about ordinary points. Section 5.4. Problems for Section 5.3.
Friday, October 13

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 16 Fall Break
Wednesday, October 18 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 20

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 23

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 25 Review for Midterm II.

Midterm II

Review Sheet.

Midterm II. 6-7 PM, 1360 East Hall. Covers Chapters 1-5.
Friday, October 27 Lecture 18: Section 6.5. Impulsive forcing. Delta functions and their Laplace transforms. Section 6.6. Problems for Section 6.5.
Week 9

Monday, October 30

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 1

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 3

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 6 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 8 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 10 Lecture 23: Section 7.6. Complex eigenvalues. Section 7.7. Problems for Section 7.6.
Week 11 Monday, November 13 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 15 Lecture 25: Section 7.8. Repeated eigenvalues and generalized eigenvectors. Section 7.9. Problems for Section 7.8.
Friday, November 17 Lecture 26: Section 7.9. Nonhomogeneous first-order linear systems. Sections 8.1-8.2. Problems for Section 7.9.
Week 12 Monday, November 20

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 22 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 24 Thanksgiving Holiday
Week 13 Monday, November 27

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 29

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 1 Lecture 30: Section 9.2. Autonomous nonlinear systems. Equilibria and stability. Section 9.3. Problems for Section 9.2.
Week 14 Monday, December 4 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 6 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 8

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 11

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 16 Tuesday, December 19 1:30-3:30 PM, 1640 CHEM

Final Exam

Review Sheet.