Math 320: Linear Algebra and Differential Equations (R. Jones)

Fall 2007

Lecturer: Dr. Raphael ("Rafe") Jones
Email: jones@math.wisc.edu
Phone: 263-5640
Office: 317 Van Vleck Hall
Office Hours:

Wednesday Thursday Friday	10:30-11:30 11:00-12:00 2:30-3:30

Or by appointment.

Math 320 lecture meeting Times: MWF 12:05–12:55PM (Van Vleck B130)

Jump to: EXAM INFORMATION Important Announcements Textbook Discussion Sections and TA Office Hours Getting Help Classlist

Important Announcements (updated Dec. 20)

Thanks to those of you who handed in your final exam note sheets. Here are some of my favorites -- scroll down to see all six.

The grand champions (Best overall, and best octopus, respectively)

The impressively organized

Cascading columns of characters

The simplicity/absurdity award

Final Exam: The final will be cumulative, with slightly more emphasis on material covered since the second exam (i.e. section 5.4 and after). As with the other exams, you'll be allowed to have one side of one sheet of paper of notes for the exam. Also, problems will be similar to problems on the HW, quizzes, and midterms.

Office Hour: I will be having an office hour for last-minute questions on Monday, 5/14 from 10:30 to noon.

For a practice final exam, click here. For solutions to the practice final, click here

There are also review problems given below.

Here is an outline of what we've covered in this class, organized by section in the book:

• What is a DE and what is a solution to a DE; How DEs arise in some simple mathematical models (1.1)
• Solving first-order DEs by integration (1.2), separation of variables (1.4), integrating factors (1.5), and substitution (1.6). Includes knowing how to recognize when a given DE is amenable to one of these techniques.
• Sketching slope fields for certain first-order DEs and knowing how to interpret them (1.3)
• Theorem on when there is a unique solution for an initial value problem of the form y' = f(x,y), y(a) = b. Even when the theorem guarantees a unique solution on some interval containing x = a, this solution can lose its uniqueness on larger intervals. (1.3)

• DEs modeling population growth: exponential and logistic populations. (2.1)
• Autonomous DEs: equilibrium solutions and their classification. Also, often you have a family of autonomous DEs depending on a parameter, and for parameter values belonging to certain intervals, the equilibrium solutions may behave differently. This information is encapsulated in a bifurcation diagram (2.2).
• Numerical approximation to solutions of certain first-order DEs using Euler's method (2.4).

• Solutions to systems of linear equations using elementary row operations (3.1 and 3.2). Includes determining whether there are 0, 1, or infinitely many solutions.
• Echelon and Reduced Echelon form for matrices (3.3)
• Arithmetic with matrices, including addition, multiplication (3.4), and inverses (3.5)
• Determinants. Their definition, how to compute them with row operations, their properties (e.g. determinant nonzero means the matrix is invertible, its columns are linearly independent, etc.) (3.6)
• Curve Fitting. We did an example showing that we can fit a polynomial curve to data points by solving a system of equations. (3.7)

• Vector Spaces, Subspaces, Linear Independence, Spans, Bases, and Dimension (see review sheet) (4.1 - 4.4)

• Second-order linear DEs, their solution sets (two-dimensional vector spaces), existence and uniqueness of solutions, linear independence of functions (Wronskians), how to solve second-order linear DEs that are homogeneous with constant coefficients (factor the characteristic equation). (5.1)
• Higher-order linear DEs. Same ideas as 5.1, except now we show that the solution sets are n-dimensional, so that finding n linearly independent solutions to the DE is enough to determine the general solution. (5.2)
• Higher-order linear DEs that are homogeneous with constant coefficients. Similar to solving second-order ones: write down the characteristic equation and factor it, then use roots to write down solutions. (5.3)
• Mechanical vibrations. Equations for motion of a spring with resistance. How to solve the resulting DEs and find important properties of the resulting motion (e.g. amplitude, period) (5.4)
• Non-homogeneous higher-order linear DEs. The general solution is y_c + y_p, where y_c is the general solution of the associated homogeneous DE, and y_p is any single solution of the non-homogenous DE. To find y_p, use the method of undetermined coefficients, or variation of parameters. (5.5)
• Forced motion and resonance. Similar to 5.4, except now there may be an outside force acting on the spring system, making the resulting DE non-homogeneous. Use the methods of 5.5 to solve these. When x_c and x_p have the same period, the result is resonance, which can be disastrous. (5.6)

• Eigenvalues and eigenvectors of matrices: what they are, and how to find them. (6.1)
• Systems of first-order linear DEs. What they are, how to write them in terms of vector-valued functions and a matrix. Solution space of an n by n system has dimension n, so only need to find n linearly independent solutions. Can check linear independence using Wronskians. (7.2)
• The eigenvalue method for homogeneous systems of linear DEs: case of no repeated eigenvalues. Find the eigenvalues of the matrix involved in the system of DEs, and then find an eigenvector associated to each one. Build n linearly independent solutions out of them (see formula on p. 421 in case of complex eigenvalues). Tank problem example. (7.3)
• The eigenvalue method for homogeneous systems of linear DEs: case of repeated eigenvalues where there is no defect. Write bases for all the eigenspaces, and then use each basis element along with its corresponding eigenvector to construct your n linearly independent solutions. (7.5)
• Non-homogeneous systems. General solutions are x_c + x_p. To find x_p, use the method of undetermined coefficients. (8.2) noting that in the case of overlap with x_c, you have to include more terms than in the case of a single DE. (8.2)

Review Problems

The following are some problems from the book that cover much of what we've done this semester. Try to do them as if they were exam questions, i.e., without looking at surrounding material in the book (or the answers in the back). If you get truly stuck, then by all means consult the material in the relevant section and/or the solution.

• 1.3 #20
• p. 76 Review Problems #6,7
• 2.1 #21
• 2.2 #9
• 2.4 #2
• 3.2 #14, 15
• 3.4 #9
• 3.5 #20
• 4.2 #11
• 4.3 #13, 20
• 4.4 #2, 3, 13
• 5.2 #9
• 5.3 #10, 23
• 5.4 #13
• 5.5 #9
• 5.6 #19
• 7.2 #15
• 7.3 #3, 14
• 8.2 #6

  Exam 2 solns now available:

  • Problem 1
  • Problem 2
  • Problem 3a
  • Problem 3b
  • Problem 3c
  • Problem 3d
  • Problem 4
  • Problem 5

  In class on 11/30 we discussed the phenomenon of resonance. This happens when a system of coupled oscillators (i.e. cart-and-flywheel system) has DE of the form mx'' + kx = F cos(wt), where the period w of the external force is the same as the period w_0 of the motion in the complementary solution x_c (i.e. solution of the DE mx'' + kx = 0). In this case our initial guess for x_p overlaps with x_c, so we have to amend x_p by multiplying by t. The presence of this t in the equation of motion for the system leads to oscillations of increasing amplitude. There is a long history of mechanical systems (such as bridges) that have been subject to periodic external forces (like a gusty wind or the rhythm of marching soldiers) whose frequencies have matched internal frequencies of the system, and the resulting resonance has been catastrophic. The book lists some on p. 352. A famous example is that of the Tacoma Narrows bridge, which had already earned the nickname "Galloping Gertie" because of how it was prone to move in the wind. It didn't gallop for long, as it was destroyed by a nearly-periodic wind with just the right frequency to create resonance See this youtube clip for footage of the disaster.

  The second exam will cover sections 3.5 to 5.3. Note that we skipped most of 3.7 (you only have to know how to do problems like example 1, p. 215) and also sections 4.5 to 4.8.

  For a reivew of the linear algebra concepts we have covered (sections 4.1 to 4.4), click here.

  Like the first exam, you'll be allowed one side of one sheet of notes, which must be handwritten by YOU (so don't use a friend's). No magnifying devices are allowed, so write in a size and style that are useful to you.

  Calculators will be neither permitted nor necessary on the exam. You can find a practice exam here. For the version with solutions, click here.

  .

  Exam 1 solns now available:

  • Problem 1
  • Problem 2a,b
  • Problem 2c,d
  • Problem 3
  • Problem 4
  • Problem 5
  • Problem 6
  • Problem 7

  There will be no class Weds, 11/21 (the Weds before Thanksgiving).

  The first exam will cover material up through 3.4. You'll be allowed one side of one sheet of notes, which must be handwritten by YOU (so don't use a friend's). No magnifying devices are allowed, so write in a size and style that are useful to you.

  Calculators will be neither permitted nor necessary on the exam. You can find a practice exam here. For the version with solutions, click here.

  .

  The homework originally due Friday, October 5 (including webwork) is now due Monday, October 8.

  Welcome to the class!

  Check out the course syllabus -- chock full of useful and inspiring info -- here.

  The first homework is due on 9/14, while the first quiz is the week of 9/17.

  Visit the course webwork site. Your login is the same as your UW login, and your password is your student ID number (ten digits). There is also a webwork help site. The first assignment is now available. It's due at 5 pm next Friday, 9/14.

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  Textbook
  	Edwards and Penney, Differential Equations and Linear Algebra, 2nd ed., Prentice Hall.
  This book is available for purchase at the University Bookstore and at the Underground Textbook Exchange, but you can compare new and used prices at several different online booksellers at CampusBooks4Less.com. Warning: If you choose to purchase the (optional) Student Solutions Manual, you may develop a syndrome known as over-reliance, which puts you at a high risk for over-confidence and, if left untreated, could result in severe exam underperformance. Use sparingly.

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  Discussion Sections        Click on your TA's name to send an email.			Office Hours
  321:	T 8:50 - 9:40 AM in B215 Van Vleck	Patrick Rault	(Van Vleck 620) Tuesday 12:20 - 1:10 PM
  322:	Th 8:50 - 9:40 AM in B215 Van Vleck	Patrick Rault	Friday 1:00 - 1:50 PM
  325:	T 1:20 - 2:15 PM in 214 Ingraham	Patrick Rault	(and by appointment)
  326:	Th 1:20 - 2:15 PM in 214 Ingraham	Patrick Rault
  323:	T 9:55 - 10:45 AM in 214 Ingraham	Melanie Ruiz	(Van Vleck 101-1) Wednesday 1:00-2:00 PM
  324:	Th 9:55 - 10:45 AM in 214 Ingraham	Melanie Ruiz	Friday 1:00-3:00 PM (and by appointment)

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  Getting Help

  The first step should always be to see your TA or Dr. Jones during office hours. If you can't make our office hours, send an email to set up an appointment with one of us. You should also check out the following resources:

  • List of Private Math Tutors
  • GUTS A free student-run program covering a variety of subjects. Staffed mostly by student volunteers.

  Classlist
  An email Classlist has been created for important announcements about this course. All students enrolled in the course are automatically added to the list. Your @wisc.edu or @students.wisc.edu email address is the one that will be used for the list, as well as for all other official communication from the University, so check your email frequently. If you are not enrolled in the course, but would like to be added to the list, please email Dr. Jones.

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  ^‡From Section VI.14 of the UW Madison College of Letters and Science Handbook: "The time of a two-hour block for a class and/or the due date for a take-home examination may be changed only with the prior approval of the associate dean for Student Academic Affairs. Such changes are rare. Where a student has more than two (that is, three or more) Summary Blocks scheduled within a period of 24 hours, the instructor may, within guidelines adopted by the College faculty, reschedule a final exam for that individual student to avoid hardship."

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  SYLLABUS

  HOMEWORK

  WEBWORK

  QUIZZES

  CLASS SCHEDULE

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  The URL for this page is  http://www.math.wisc.edu/~jones/Math320
  

  jones@math.wisc.edu