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

Spring 2006

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

Wednesday Thursday Friday	11:00-12:00 1:30-2:30 2:30-3:30

Or by appointment.

Math 320 lecture meeting Times: MWF 9:55–10:45AM (Bascom 165)

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

Important Announcements (updated May 13)

Final Exam: The final will be cumulative, with slightly more emphasis on material covered since the second exam (i.e. section 5.5 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 HW and quiz problems.

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

Exam 2 solutions can be found here.

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), and integrating factors (1.5). 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) and the improved Euler's method (2.5).

• 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 go back and forth between systems of first-order DEs and single higher-order DEs. Writing a first-order system 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. (7.1, 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. Construct chains of generalized enigenvectors and build linearly independent solutions out of them. When the eigenvalue has defect 2, one method of doing this is to find v_2 such that (A - lambda I)^2 v_2 = 0 and (A - lambda I) v_2 is not zero. Then, putting v_1 = (A - lambda I) v_2, the generalized eigenvectors are v_1 and v_2. (7.5)
• Non-homogeneous systems. General solutions are x_c + x_p. To find x_p, use the method of undetermined coefficients, 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
For a handout of review problems, click here.

The following are some problems from the book that cover much of what we've done this semester.

• 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
• 7.5 #5
• 8.2 #6

  
  
  Brief recap of class on 4/20: We discussed much of section 5.6, focusing on DEs of the form mx'' + cx' + kx = F cos(wt), which correspond to a mass-flywheel system (see picture on p. 348). We solved this DE and saw that there are two very different kinds of behavior, depending on whether w = w_0, where w_0 is the period of the motion in the complementary solution x_c of the DE mx'' + cx' + kx = 0. When w is different from w_0, one gets a "mixed" kind of double oscillation (see Example 1 and picture on p. 350). When w = w_0 then the particular solution x_p is of the form tsin(w_0t), and this has amplitude that grows infinitely large as t goes to infinity, thereby destroying the system! This phenomenon is known as resonance, and there is a long history of its occurrences (see discussion on p. 352). One of the most spectacular instances of resonance happened when the Tacoma Narrows bridge, which had already earned the nickname "Galloping Gertie" because of how it was prone to move in the wind, was destroyed by a nearly-periodic wind with just the right frequency to create resonance (See these amazing youtube clips; the third clip is the one I would recommend). In class, we did the examples of x'' + 4x = cos(3t), where there is no resonance, and x'' + 4x = cos(2t), where there is resonance.

  Practice exam for exam 2 now posted.

  Solutions to the first exam now available.

  Note on HW #2: for problems 13, 14, and 20 of section 1.3, you should first determine whether Theorem 1 guarantees that there is a unique solution to the given initial value problem. If it does not, determine whether there is no solution to the initial value problem, or more than one.

  Welcome to the class!

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

  Also visit the course webwork site, where the second HW assignment is now available (due Friday 5pm). Your login is the same as your UW login, and your password is your student ID number (ten digits).

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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
  301:	T 8:50 - 9:40 AM in B333 Van Vleck	Michael Woodbury	Tuesday 3:45-4:45
  302:	Th 8:50 - 9:40 AM in B333 Van Vleck	Michael Woodbury	Thursday 12:30-1:30
  303:	T 9:55 - 10:45 AM in B325 Van Vleck	Michael Woodbury	Friday 1:30-2:30
  304:	Th 9:55 - 10:45 AM in B325 Van Vleck	Michael Woodbury

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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