Past Announcements:

Available on a separate announcement page.

Latest Announcement (Thursday, May 12): Final Exam, 8 AM

Dear all,

A final reminder that your final exam is 8 AM - 10:45 AM in the usual classroom. Please make sure to come prepared with the usual exam accoutrements (SB ID, pencils, pens, erasers). See you bright and early!

(And fill out course evaluations!)

Best,

Kevin

Available via Google Drive.

TA: Junbang Liu

Office Hours: W, 1:00-2:00 PM, Zoom

MLC Hours: W, 2:00-4:00 PM, Zoom

R02

Time and Location: W 4:25-5:20 PM, Physics P-130

R04

Time and Location: Tu 6:30-7:25 PM, Earth & Space 181

Required

Differential Equations and Boundary Value Problems: Computing and Modeling (5th Edition) by Edwards, Penney, and Calvis

• We will be largely following this textbook, and homework problems will be assigned from it.

Complementary

Dr. Trefor Bazett's YouTube series on Ordinary Differential Equations

• Video lectures covering much of the same material we will cover in class.

MIT OpenCourseWare 18.03SC from Fall 2011

• Contains all sorts of useful material, including video lectures and lecture notes, from a course similar to ours.

Supplementary

Grant Sanderson's (3Blue1Brown's) YouTube series on Differential Equations

• Grant is a master expositor. His YouTube series delves into much more theory than we will see, including partial differential equations, but his videos are so good that even if you haven't seen multivariable calculus, you will get something out of them.

From the Undergraduate Bulletin: Homogeneous and inhomogeneous linear differential equations; systems of linear differential equations; series solutions; Laplace transforms; Fourier series. Applications to economics, engineering, and all sciences with emphasis on numerical and graphical solutions; use of computers.

Instructor's Description: In nature, one often finds that certain physical quantities which change over time have that the rate of change is related to the value of the quantity at a certain time. For example, think of the population of the world. Naively, the greater the population of a community, the faster that the population will increase, at a rate proportional to the population. (This assumes there is no limit to resources, and hence no worry about overpopulation.) Suppose that we write P for the population as a function of time t. Then the rate of change of a population is just dP/dt, and so our naïve model for population increase has a mathematical description by the equation

dP/dt = cP

for some constant c > 0. This is an example of a (ordinary) differential equation, relating the derivative of the function P to its value. More general differential equations can also involve higher derivatives, and also the value of t.

The goal of this course is to take differential equations, like the one above for population, and attempt to find which functions P satisfy the equation. In our example, given that dP/dt = cP, what are the possibilities for P? It turns out that the only possibility is an exponential function of the form P(t) = Ae^(ct), for some constant A (and where c is the same constant appearing in the differential equation). In other words, simply from our assumption that population grows at a rate proportional to the population itself, we find that the population grows exponentially.

This type of differential equation is of the simplest possible type. Nature is rarely so simple, however. So we will need to be able to solve much more complicated differential equations, for which we will develop a number of tools. We will stay within Chapters 1-6 of the textbook Differential Equations and Boundary Value Problems (5th Edition) by Edwards, Penney, and Calvis.

Below is a tentative schedule for lectures, where the section numbers refer to the textbook.

Lecture 1 (M, Jan 24): (SLIDES) Logistics, 1.1 Differential Equations and Mathematical Models

Lecture 2 (W, Jan 26): (SLIDES) 1.1 Differential Equations and Mathematical Models, 1.2 Integrals as General and Particular Solutions

Lecture 3 (F, Jan 28): (SLIDES) 1.3 Slope Fields and Solution Curves

Lecture 4 (M, Jan 31): (SLIDES) 1.4 Separable Equations and Applications

Lecture 5 (W, Feb 2): (SLIDES) 1.4 Separable Equations and Applications, 1.5 Linear First-Order Equations

Lecture 6 (F, Feb 4): (SLIDES) 1.5 Linear First-Order Equations

Lecture 7 (M, Feb 7): (SLIDES (Part 1), NOTES (Part 2)) 1.5 Linear First-Order Equations, 1.6 Substitution Methods and Exact Equations

Lecture 8 (W, Feb 9): (NOTES) QUIZ 1 (Sections 1.1-1.4), 1.6 Substitution Methods and Exact Equations

Lecture 9 (F, Feb 11): (NOTES) 1.6 Substitution Methods and Exact Equations

Lecture 10 (M, Feb 14): (NOTES) 1.6 Substitution Methods and Exact Equations

Lecture 11 (W, Feb 16): (NOTES) 1.6 Substitution Methods and Exact Equations, 2.1 Population Models

Lecture 12 (F, Feb 18): (NOTES) 2.1 Population Models, 2.2 Equilibrium Solutions and Stability

Lecture 13 (M, Feb 21): (NOTES) 2.2 Equilibrium Solutions and Stability

Lecture 14 (W, Feb 23): (NOTES) QUIZ 2 (Sections 1.5-1.6), 2.2 Equilibrium Solutions and Stability, 2.3 Acceleration-Velocity Models

Lecture 15 (F, Feb 25): (NOTES) 2.3 Acceleration-Velocity Models, 2.4 Numerical Approximation: Euler's Method

Lecture 16 (M, Feb 28): (NOTES) 2.4 Numerical Approximation: Euler's Method

Lecture 17 (W, Mar 2): (NOTES) 3.1 Introduction: Second-Order Linear Equations

Lecture 18 (F, Mar 4): (NOTES) 3.1 Introduction: Second-Order Linear Equations, 3.2 General Solutions of Linear Equations

Lecture 19 (M, Mar 7): (NOTES) Review, 3.2 General Solutions of Linear Equations

MIDTERM!!! (W, Mar 9): (Covers Material up to Lecture 16 (Sections 1.1-1.6, 2.1-2.4)

Lecture 20 (F, Mar 11): (NOTES) 3.2 General Solutions of Linear Equations

SPRING BREAK!!!!

Lecture 21 (M, Mar 21): (NOTES) 3.2 General Solutions of Linear Equations, 3.3 Homogeneous Equations with Constant Coefficients

Lecture 22 (W, Mar 23): (NOTES) 3.3 Homogeneous Equations with Constant Coefficients

Lecture 23 (F, Mar 25): (NOTES) 3.3 Homogeneous Equations with Constant Coefficients, 3.4 Mechanical Vibrations

Lecture 24 (M, Mar 28): (NOTES) 3.4 Mechanical Vibrations

Lecture 25 (W, Mar 30): (NOTES) 3.5 Nonhomogeneous Equations and Undetermined Coefficients

Lecture 26 (F, Apr 1): (NOTES) 3.5 Nonhomogeneous Equations and Undetermined Coefficients

Lecture 27 (M, Apr 4): (NOTES, Variation of Parameters Justification) 3.5 Nonhomogeneous Equations and Undetermined Coefficients

Lecture 28 (W, Apr 6): (NOTES) QUIZ 3 (Sections 3.1-3.4), 4.1 First Order Systems and Applications

Lecture 29 (F, Apr 8): (NOTES, Mathematica 2-D Direction Field / Phase Portrait) 4.1 First Order Systems and Applications

Lecture 30 (M, Apr 11): (NOTES) 4.1 First Order Systems and Applications, 4.2 The Method of Elimination

Lecture 31 (W, Apr 13): (NOTES) 4.2 The Method of Elimination

Lecture 32 (F, Apr 15): (NOTES) 5.1 Matrices and Linear Systems

Lecture 33 (M, Apr 18): (NOTES) 5.1 Matrices and Linear Systems

Lecture 34 (W, Apr 20): (NOTES) 5.1 Matrices and Linear Systems

Lecture 35 (F, Apr 22): (NOTES) 5.2 The Eigenvalue Method for Homogeneous Systems

Lecture 36 (M, Apr 25): (NOTES, Mathematica 2-D linear systems) More on row reduction, 5.2 The Eigenvalue Method for Homogeneous Systems

Lecture 37 (W, Apr 27): (NOTES) QUIZ 4 (Sections 4.1, 4.2, 5.1), 5.2 The Eigenvalue Method for Homogeneous Systems

Lecture 38 (F, Apr 29): (NOTES) 5.5 Multiple Eigenvalue Solutions

Lecture 39 (M, May 2): (NOTES) 5.5 Multiple Eigenvalue Solutions

Lecture 40 (W, May 4): (NOTES) 5.5 Multiple Eigenvalue Solutions

Lecture 41 (F, May 6): Discussion of the "further reading" (non-examinable, no notes)

Further Reading (non-examinable): 5.3, 5.4, 5.6, 5.7, 6.1

• Section 5.3 provides intuition for how the eigenvectors and eigenvalues of the matrix in a constant coefficient homogeneous first order linear system appear in the direction field / phase portrait of the system.

• Section 5.4 gives physical applications of the constant coefficient homogeneous first order linear systems.

• Section 5.6 gives a general understanding of the full theory of constant coefficient homogeneous first order linear systems in terms of matrix exponentials.

• Section 5.7 treats the nonhomogeneous case for constant coefficient first order linear systems.

• Section 6.1 describes how for a non-linear system, one can "linearize" it so that the analysis of Chapter 5 applies on the small scale, i.e. "near" the equilibria. This section, in my opinion, is a massive punchline for the course that unfortunately appears *just* outside of the syllabus.

FINAL EXAM!!!! (F, May 13) Same room, but 8:00-10:45 AM

There will be four short in-class quizzes (two questions, about 15-20 minutes), to occur at the beginning of class (9:15-9:30 or 9:35 AM) on Wednesdays throughout the semester. They serve the purpose of assessment for the instructors as well as self-assessment for the students.

• Quiz 1: W, Feb 9 (15 minutes)

  • Quiz 1 + Solutions

  • Statistics

    • Grade distribution and approximate letter grades

    • Mean = 18.88 (out of 20), Median = 20 (out of 20)

• Quiz 2: W, Feb 23

  • Quiz 2 + Solutions

  • Statistics

    • Grade distribution and approximate letter grades

    • Mean = 15.23 (out of 20), Median = 17 (out of 20)

• Quiz 3: W, Apr 6

  • Quiz 3 + Solutions

  • Statistics:

    • Grade distribution and approximate letter grades

    • Mean = 15.98 (out of 20), Median = 17 (out of 20)

• Quiz 4: W, Apr 27

  • Quiz 4+ Solutions

  • Statistics:

    • Grade distribution and approximate letter grades

    • Mean = 11.51 (out of 20), Median = 10 (out of 20)

Midterm:

• Written exam, occurring in-class on W, March 9

• Review Sheet

• Practice Midterm + Solutions

• Midterm + Solutions

• Statistics

  • Grade distribution and approximate letter grades

  • Mean = 58.82 (out of 80), Median = 62 (out of 80)

Final:

• Written exam, F, May 13, 8:00-10:45 AM in the usual room (note the unusual time)

• Practice Final + Solutions

• Final + Solutions

• Statistics (not including students who didn't take the exam):

  • Grade distribution

  • Mean = 76.74 (out of 130), Median = 77 (out of 130)

The grade thresholds are as follows:

• ≥ 89% earns an A

• ≥ 86% earns an A- or A

• ≥ 83% earns a B+ or better

• ≥ 73% earns a B or better

• ≥ 68% earns a B- or better

• ≥ 64% earns a C+ or better

• ≥ 48% earns a C or better

• ≥ 44% earns a C- or better

• ≥ 39% earns a D or better