MATH 209: Differential
Equations and Vector Calculus: MWF 11 - 11:50am
Bronfman 106
Office hours: M-F 10:05-10:50am, M-F 4-4:30pm, whenever I'm in my office (click here for my schedule)
TA Review Sessions: Sunday 9-11pm in CLARK 204 and Monday 9-11pm in BRONFMAN 106
FINAL IS SATURDAY, MAY 23 AT 1:30PM IN BRONFMAN 105
Review sessions: Thurs May 14th from 1:30 - 3:30 in Bronfman 107;
Thurs May 21st from 2 - 4pm, Fri May 22nd from 10-11am and 1:30-2:30pm (all in Bronfman 104)
takeaways from the course techniques for solving differential equations
course description HW/Exams/Grading policy HW and reading Handouts/Programs Project Topics Other Links (summer research) additional comments
COURSE DESCRIPTION: Historically, much beautiful mathematics has arisen from attempts to explain heat flow, chemical reactions, biological processes, or magnetic fields. A few ingenious techniques solve a surprisingly large fraction of the associated ordinary and partial differential equations. We will start with difference equations. These are the discrete analogues of differential equations, and have numerous applications in both pure and applied math (for example, a generalization of the Fibonacci numbers explains why double-plus-one will almost surely bankrupt you if you play roulette in Las Vegas!). After studying these we’ll move on to differential equations, describing both general existence and uniqueness theorems as well as techniques to solve the systems (which range from complete solutions to numerical approximations). Examples will be drawn from pure mathematics, physics, biology, as well as from class requests. As time permits, we will describe special functions and advanced topics (possibilities include Random Matrix Theory and the Calculus of Variations). Format: lecture/discussion. Evaluation will be based on problem sets, hour tests, and a final exam. Prerequisites: Mathematics 102 (or demonstrated proficiency on a diagnostic test; see Mathematics 101). No enrollment limit (expected: 30).
HOMEWORK / EXAMS / GRADING: I encourage you to work in groups, but everyone must submit their own HW assignment. HW is to be handed in on time, stapled and neat -- late, sloppy or unstapled HW will not be graded. Please show your work on the HW and exams (otherwise you risk getting no credit). Grading will be: 20% homework, 40% midterms (there will be two), 40% final. You may also do a project involving differential equations, which would count for 10% of your grade (and the other categories would be reduced 10% each). All exams are cumulative. Click here for an example on how to write up a homework (this is the solution to the first two difference equation problems).
SYLLABUS / GENERAL: The textbook will be the 9^{th} edition of Boyce and DiPrima’s `Elementary Differential Equations and Boundary Value Problems’ (YOU MAY USE THE EIGHT EDITION FOR THIS CLASS -- IF THE HW PROBLEMS DIFFER, I WILL POST THE PROBLEM FROM THE NINTH EDITION.). We will follow the book closely, covering much of the first 8 chapters, occasionally supplementing with additional material from other sources. My lecture notes are also available online here (click here for additional comments on each lecture); note, of course, that these are rough notes to help me give each lecture and thus will not include everything mentioned in class. Also, please feel free to swing by my office or mention before, in or after class any questions or concerns you have about the course. If you have any suggestions for improvements, ranging from method of presentation to choice of examples, just let me know. If you would prefer to make these suggestions anonymously, you can send email from mathephs@gmail.com (the password is the first seven Fibonacci numbers, 11235813).
OBJECTIVES: There are two main goals to this course: to learn how to solve difference and differential equations, and to learn how to model real world problems and how to attack their solution. We will constantly emphasize the techniques we use to solve problems, as these techniques are applicable to a wide range of problems in the sciences.
HOMEWORK AND READING: We will cover the following sections (as well as others as time permits):
Chapter 1: Introduction: 1.1, 1.2, 1.3.
Chapter 2: First Order Linear Equations: We will start with difference equations (see also Section 2.9), 2.2, 2.5 (see also Tim Penning's Do Dogs Know Bifurcations), 2.6, 2.7, 2.8.
Chapter 3: Second Order Linear Equations: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7.
Chapter 4: Higher Order Linear Equations: 4.1.
Chapter 5: Series Solutions of Second Order Linear Equations: 5.1, 5.2.
Chapter 6: The Laplace Transform: 6.1, 6.2.
Chapter 8: Numerical Methods: 8.1,
Chapter 10: TBD
Calculus of Variations: TBD
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Week 1: Feb 2 to 6:
Read: Chapter 1: 1.1, 1.2, 1.3.
HW: none due
Supplemental: Study sums such as Sum_{n = 1 to oo} n / 2^n, Sum_{n = 1 to oo} n / 3^n, Sum_{n = 1 to oo} n / 4^n, .... Can you predict the general form of Sum_{n = 1 to oo} n r^n? What about sums such as Sum_{n = 1 to oo} 1 / n^2?
Week 2: Feb 9 to 13:
Read: Chapter 1: 1.1, 1.2, 1.3; Chapter 2: 2.1, 2.2.
HW: Due Wednesday, February 18: From Problems on difference equations: Exercises #1.1, #1.4; Chapter 1: 1.1: #5; 1.2: #4; #15; 1.3: #21, #22; Chapter 2: 2.1: #15, #20; 2.2: #2, #3, #30abcde (do not do f).
Supplemental: From Problems on difference equations: Exercises 1.2, 1.5, 1.6, 1.7; Chapter 1: 1.1: #32; 1.2: #32; Chapter 2: 2.1: #31, #38; 2.2: #31.
Week 3: Feb 16 to 20: (no class on Friday)
Read: Chapter 2: 2.1, 2.2,2.4, 2.5.
HW: Due Wednesday, February 25: Chapter 2: 2.1: #19, #29; 2.2: #5, #8; 2.4: #25, #27, #28; 2.5: #3, #18, #28; 2.6: #4, #18. Solve dy/dx = (2x+5y) / (3x+4y).
Supplemental: Chapter 2: 2.1: #33; 2.2: #29; 2.4: #26, #31, #32; 2.5: #16, #22-#24 (disease models), #25 (bifurcations); 2.6: #18, #19.
Week 4: Feb 23 to 27:
Read: Chapter 2: 2.5, 2.6, 2.7, 2.8, 3.1.
HW: Due Wednesday, March 4: Chapter 2: 2.7: #1ad (ie, only do step size h=.1); 2.8: #1, #7 (do part (a), where by an arbitrary value of n the book means ANY value of n, not an n of your choice; further, if possible solve the differential equation using the techniques from this chapter or show why none of our methods work); Miscellaneous problems (page 132 in 9th edition, 131 in 8th): #1, #9, #22. Chapter 3: 3.1: #18, #20, #23. Construct a sequence of continuous functions f_n(x) converging to a continuous function f(x) such that there is an M so that, for all x and n, |f_n(x)| and |f(x)| are at most M but lim_{n -> oo} Int_{-oo to oo} f_n(x)dx does not equal Int_{-oo to oo} lim_{n -> oo} f_n(x)dx.
Extra credit. Consider the difference equation x_{n+1} = (x_n + n * x_{n-1}) / (n+1). (1) If x_0 = 0 and x_1 = 1: (i) prove that lim_{n --> oo} x_n exists and (ii) calculate this limit. (2) If now x_0 = a and x_1 = b, with a and b any real numbers: (i) prove that lim_{n --> oo} x_n exists and (ii) calculate this limit.
Supplemental: Chapter 2: 2.7: #20; 2.8: #13, #14, #19, #36, #42. Chapter 3: 3.1: #17, #28.
Week 5: March 2 to 6:
Read: Chapter 3: 3.2, 3.3, 3.4, 3.5, 3.6, 3.7. NOTE: Section 3.3 of Chapter 3 isn't in the 9th edition; we will only briefly cover this material.
Linear algebra review: (1) multiplying matrices (html, word); (2) many notes (you do not need the many notes for this course; this is a resource for anyone interested)
HW: Due Wednesday, March 11: 3.2: #5, #7, #14, and do #22 in the 9th edition of the textbook (which is #21 in the 8th edition); Three other problems: (1) Find the general solution. to y'' - 2y' + 2y = 0. (2) Consider 5u'' + 2u' + 7u = 0 with u(0) = 2 and u'(0) = 0. Find the solution u(t), and deterime for t > 0 the first time when |u(t)| = 10; (3) Solve y'' - 2y' + y = 0; (4) If the roots of the characteristic equation are real, show that a solution of ay'' + by' + cy = 0 is either everywhere zero or it is zero at most once.
Supplemental: 3.2: #2, #15, any question from the Exact or Adjoint Equations sections; If you have the 9th edition, do #34 of Section 3.3, and if you have the 8th edition do #38 of Section 3.4;
Wronskians and Variation of Parameters in pop culture: http://www.youtube.com/watch?v=fdDL4dFb-VY&feature=related
Week 6: March 9 to 13: Midterm will be Friday, March 20th.
Read: Chapter 3: 3.5, 3.6; Chapter 4: 4.1.
HW: Due Wednesday, March 18: (1) Solve y'' - 2y' - 3y = 3 exp(2t); (2) y'' + 2y' + 5y = 3sin(2t); (3) solve the following by variation of parameters AND by undetermined coefficients: y'' - 5y' + 6y = 2 exp(t); (4) Show y_1(t) = t^2 and y_2(t) = 1/t solve t^2 y'' - 2y = 0, and then find all solutions to the differential equation t^2 y'' - 2y = 3t^2 - 1; Section 3.7: #28 and #29. Also do: (5) Find all solutions to y'''' - 5y'' + 4y = 0; (6) Find all solutions to y''' - 4y'' + 5y - 2 = 0, then choose any set of initial conditions that uniquely determine a solution and find that solution.
Supplemental: 3.6: #22, #23, #24, #25, #26, #27. 4.1: #26, #27.
Week 7: March 16 to 20: Midterm will be Friday, March 20th.
Read: Chapter 5: 5.1, 5.2
Read about sequences and series and their convergence; see also the links from Wednesday, March 18th.
HW: Due Wednesday, April 8: 5.1: #1, #2, #7, #8, #9, #13, #18.
Supplemental: 5.1: #6, #19,
Week 8: March 23 to 27: SPRING BREAK
Read: Chapter 7: 7.1, 7.2. You can also look at my lecture notes just for chapter 7. Chapter 7 requires linear algebra; we will do just enough linear algebra to see the beauty and power of the theory WITHOUT getting bogged down in the linear algebra complications. Thus, while we will be following chapter 7, there is a lot of material in chapter 7 that we will NOT do as this course does not have linear algebra as a pre-requisite. You are responsible only for what we do in class, NOT what is in the book; however, the book is a good reference and has additional remarks and more theory for those who are interested.
HW:
Supplemental:
Week 9: March 30 to April 3rd: SPRING BREAK
Read: Chapter 7: 7.1, 7.2. You can also look at my lecture notes just for chapter 7. Chapter 7 requires linear algebra; we will do just enough linear algebra to see the beauty and power of the theory WITHOUT getting bogged down in the linear algebra complications. Thus, while we will be following chapter 7, there is a lot of material in chapter 7 that we will NOT do as this course does not have linear algebra as a pre-requisite. You are responsible only for what we do in class, NOT what is in the book; however, the book is a good reference and has additional remarks and more theory for those who are interested.
HW:
Supplemental:
Week 10: April 6 to 10:
Read: Chapter 7: 7.1, 7.2, 7.3, 7.4, 7.5. You should also look at my lecture notes just for chapter 7. Chapter 7 requires linear algebra; we will do just enough linear algebra to see the beauty and power of the theory WITHOUT getting bogged down in the linear algebra complications. Thus, while we will be following chapter 7, there is a lot of material in chapter 7 that we will NOT do as this course does not have linear algebra as a pre-requisite. You are responsible only for what we do in class, NOT what is in the book; however, the book is a good reference and has additional remarks and more theory for those who are interested.
Programs:
HW: Due Wednesday, April 15: Chapter 5, Section 5.2: Find power series solutions for the following differential equations (this will involve determining the recurrence relations and, if possible, formulas for the general term in the series expansion), and compute the Wronskian of the two solutions to show they are independent: (1) (1-x)y'' + y = 0 about the point x_0 = 0; (2) xy'' + y' + xy = 0 about the point x_0 = 1; do also Section 5.2, #21 (do all of part (a), and for (b) just do lambda = 0, 4 and 10). Chapter 7, Section 7.1: #1, #15. Section 7.2: Let A be the matrix with first row (1,3) and second row (3,1); let B be the matrix with first row (2,4) and second row (3,6). Find AB, BA and the eigenvalues of A and the eigenvalues of B. Note the two matrices are: General question: Write down a model involving either difference or differential equations for anything. You do not have to worry about whether or not your model is solvable, but try to write down a realistic model for something you find interesting. Thus, in class we wrote down a simple model for a battle between red and blue forces; this model is a bit naive but it has some nice features (primarily it is solvable!).
Supplemental: Read sections 5.3 and 5.4 and find series solutions to some of these problems.
Week 11: April 13 to 17:
HW: Due April 22nd: Do the problems in the handout linked here.
Supplemental:
Week 12: April 20 to 24: Optional midterm in class on Friday, May 1.
Read: Keep reading the online notes for chapter 7. Read my notes for Chapter 8, Section 8.1 and Section 8.1 in the book. My notes on Simpson's rule and Euler's method are here
HW: Due April 29th: Do the problems in the handout linked here.
Supplemental:
Week 13: April 27 to May 1: Optional midterm in class on Friday, May 1.
Read: Below are two papers (as well as slides from a talk) on
modeling (in baseball and in marketing), as well as notes from the class
lecture:
HW: Due May 6th: Now that we have had two lectures on mathematical modeling, write down a model for some system you care about. If possible, solve it (or describe why you cannot solve it). Describe what features of your model you feel are reasonable and what are not (but have been chosen to get something tractable, where hopefully the errors cancel out). This assignment will be graded as done / not done; however, if you find your system interesting and want to investigate it further, let me know.
Supplemental: Mathematica simulation of modeling whale evolution with products of random matrices.
Week May 4 to 8:
Read: Pages 22 and 23 of my lecture notes (on Laplace Transform) and Chapter 6, Sections 6.1 and 6.2. There will be reading (TBD) about Chaos Theory for Friday's class.
HW: Due May 13th: do the problems from the handout linked here.
Supplemental: Mathematica simulation of modeling whale evolution with products of random matrices (click here for a .pdf in case you don't have a Mathematica reader).
Week 11 to 15: Friday is the last day of classes; Review sessions: Thurs May 14th from 1:30 - 3:30 in Bronfman 107
Week 18 to 23: Final exam: Saturday, May 23rd at 1:30pm in Bronfman 105; Review Sessions: Thurs May 21st from 2 - 4pm, Fri May 22nd from 10-11am and 1:30-2:30pm (all in Bronfman 104)
Read:
HW:
Supplemental:
My lecture notes are available online here (additional notes: my notes on Simpson's rule and Euler's method are here)
linear algebra notes: (1) multiplying matrices (html, word); (2) many notes
Do Dogs Know Bifurcations (Tim Pennings)
Economics examples:
Solow growth model (separable equation): Warren Wessecker's notes, Chris Edmond's notes
MATHEMATICA PROGRAMS:
SUGGESTIONS FOR PROJECT TOPICS: Below are some possible topics; this is by no means a complete list, but rather some suggestions. Most of the links are to wikipedia to get you started; it is also worth seeing the topics listed in their differential equations section.
Quantum Mechanics ([X,P] ~ ihI/2pi, the Schodinger equation, ...)
Random Matrix Theory: GUE/GOE/GSE
Disease propagation in graphs (see me -- this is a project I'm currently working on)
Laplace equation (Dirichlet or Neuman problem or the Cauchy boundary condition)
Series solutions (see the textbook)
Lorenz equations (this is one example of chaos)
Proofs of big theorems (see the textbook for references; needed inputs can include the implicit function theorem, ...)
Three body problem (classical mechanics: three point masses under gravity; this is one of the earliest example of chaos)
Reaction rates (chemistry)
Numerical approximations (Runge-Kutta, Simpson's rule, ...)
Alderson Drive (I'm trying to get the equations, but if anyone can get them....)
Numerical approximations (Runge-Kutta, Simpson's rule, ...)
Alderson
Drive (I'm trying to get the equations, but if anyone can get them....)
- Summer Research:
- SMALL (at Williams, deadline Feb 11)
- AMS, MAA and NSF links to summer research programs.
- The Green Chicken / Math Puzzle Night
course description HW/Exams/Grading policy HW and reading Handouts/Programs Project Topics Other Links (summer research) additional comments