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Mathematics 51, Fall 2012


ARC review sessions in Braker Hall 001:(Double-check these against the
ARC schedule to be sure.)

If you are requesting an accommodation as a result of a documented disability, you must register with the Disability Services Office at the beginning of the semester. To do so, please call the Student Service Desk at 617-627-2000 to arrange an appointment with the Program Director of Disability Services. Your instructor does not need to be apprised of such needs.
Students needing any special arrangements for exams due to scheduling conflicts or because of accommodations arranged by the Program Director of Disability Services should contact well in advance, preferably in the first week of the semester.


  • The syllabus.
  • Archived syllabi for Spring 2012 Fall 2011 Spring 2011 Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007.
  • The Fall 2006 syllabus was ordered differently, so the exams do not quite correspond.
  • Sections:
    1. Hasselblatt (B block)
    2. Adler (C block)
    3. Barthelmé (F block)
  • Examination dates and grading policies are here
    At the beginning of the semester, please check these dates against any other examinations you have scheduled.
    If there are any time conflicts, please notify immediately.
  • Homework: Homework is due at the beginning of the next class.
    You are encouraged to collaborate, but you must hand in solutions written in your own hand.
    Please mark your homework (and folder, if used in the course) with the course and section numbers as well as an identifier to help you know that it is yours---something that is likely unique to your section and something that is pronounceable in case your instructor chooses to return homework by calling out the identifiers. Please write it as clearly as possible and make sure to tell your instructor well before the end of the semester what your identifier is, so credit associated with it can be counted towards your course grade.
    Feel free to use your name as your identifier, but since homework is handed off between instructor and grader in a way that does not ensure confidentiality (such as by way of drawers in the lobby of the Bromfield-Pearson building), putting your name on the homework or the folder means that you opt out of being guaranteed the confidentiality of this part of your course work.

    You receive one point if your homework contains Do not claim credit for any parts of solutions copied from the blackboard during class!
    Your homework credit is \(\displaystyle H=u_{26}(n)\cdot\frac{n-20}4\), where \(n\) is the number of homework points and \(u\) is as on p. 449 of the text.
  • Textbook

  • Textbook: Reprint of M. M. Guterman, Z. H. Nitecki, Differential Equations -- A First Course, 3rd ed., Saunders (1992).
    Saunders ISBN 0-03-072878-9, reprint ISBN 81-89617-20-6
    The Tufts bookstore web site can create the impression that there is a 6th edition. There is not.
    The reprint is good enough for the course.
    If you can get a used copy of the originally published book (the cover looks like the background of this web page), even better.
  • Where to buy the text inexpensively? Some fellow students have suggested Amazon or, and AddAll seems to be a another good start (no endorsement implied, and note that you won't get the same return policy as from the Tufts bookstore!). There are many other options, so do look around.
    The ISBN need not match exactly, but make sure that the authors, title and edition match. There are earlier editions, and the same authors have another book with a similar but different title.
  • The course uses portions of the book that are not included in the reprint (but can be found in the original Saunders book). These are
  • Chapter 1 (plus 2A) of the text - for those whose book did not arrive on time.
  • A supplement to the text that introduces complex numbers and how to do partial fractions, undetermined coefficients and Laplace transforms using complex numbers systematically.
  • Office hours, tutoring, review sessions

    The purpose of going to office hours and to obtain tutoring is to deepen your understanding of the subject matter of a course. If you are doing well in this class, tutoring can help be an outstanding student in it. In high school this is an expensive add-on, and here it is a free service. The ARC has excellent tutors available.
  • Utilize the office hours offered by the instructor of your section. We are happy to see you, we'll get to know you better, and during office hours you may get a perspective on the subject that you did not see in class.
  • During reading period there are special office hours
  • The ARC drop-in hours are to be determined; last semester they were as follows:
    Sundays7:00 - 9:00pmTilton Hall Study Lounge
    Tuesdays3:00 - 5:00pmCampus Center 209
    4:30 - 6:30pmHill Hall 2-31 / 2-32
    Wednesdays5:30 - 8:00 pmCampus Center 209
    Thursdays3:00 - 5:00pmHouston Hall 218
    4:00 - 7:00pmCampus Center 209
  • Find a tutor by checking the schedule or as follows:
    1. Log onto student web center using your Student ID Number and password.
    2. From the Navigation Menu, choose Tutor Finder.
    3. Choose View Available Tutoring Subjects.
    4. Click the Subject, followed by the course number to find the tutors available in that subject area.
    5. You will see all of the available hours for tutors in that subject.
    6. Click "Reserve" to reserve a one-on-one session with a tutor. Or click "Notify" to let a tutor know you plan on attending their drop-in hours.
    7. Fill out the information in the form.
    8. The Online Tutor Finder will immediately email the tutor and you with a confirmation of the tutoring appointment (subject, date, time, name, location)
    9. If you need to cancel, you can do so online up to 6 hours before the appointment.
  • ARC review sessions are scheduled for September 30th, 12-2pm, October 28th, 12-2pm, December 2nd, 5-7pm, and December 12th, 4-6pm, all in Braker Hall 001.
  • Study aids and supplements to the text

  • A supplement to the text that introduces complex numbers and how to do partial fractions, undetermined coefficients and Laplace transforms using complex numbers systematically.
  • Table of Laplace transform formulas.
    This will be provided for the second exam and the final exam. Feel free to use it for homework as well.
  • Free online lectures
  • Examples of making it easier to find eigenvalues. These use the properties of determinants summarized in Note 2. on page 192f of the book — review these!
  • Help with
  • Summary of spring dynamics
  • The Vandermonde determinant (useful for Exercise 2.4.13a, but Exercise 2.5.26 provides a more interesting idea)
  • Simplifying characteristic polynomials by row or column operations
  • Summary of the row reduction operations
  • Reading about Euler's method: Lamar University and Wikipedia and a demo at Cal State Fullerton about increasing the number of steps.
  • Answers to even-numbered review problems in Chapter 3
  • Worked solutions for Exercise 1 on page 321, Exercise 2 on page 321, Exercise 8 on page 322.
  • Answers to even-numbered exercises on page 321f.
  • Phase portraits of linear systems.
  • Notes on phase portraits of nonlinear systems.
  • Assistance for phase portraits of nonlinear systems from Rutgers (very sensitive to spaces and the like in the input field!!!!).
  • Suggestions for exam preparation:
  • Old exams

    An online collection of exams is at the
    Student Services website.
    It's a secure site - log in as you would for SIS Online and choose the online exams option once you're logged in.
  • Exam 1, Spring 1998Solutions
  • Exam 2, Fall 2000 There is a misprint on problem 3: \(t^2D^2\) should be \(tD^2\). — Solutions
  • Exam 2, Spring 2001Solutions
  • Exam 1, Fall 2006, including solutions
  • Exam 2, Fall 2006, including solutions
  • Exam 3, Fall 2006, including solutions
  • Final examination, Fall 2006Solutions
  • Exam 1, Spring 2007Solutions
  • Exam 2, Spring 2007Solutions
  • Exam 3, Spring 2007Solutions
  • Final Exam, Spring 2007Solutions
  • Exam 1, Fall 2007Solutions
  • Exam 2, Fall 2007Solutions
  • Exam 3, Fall 2007Solutions
  • Final Exam, Fall 2007Solutions
  • Exam 1, Spring 2008Solutions
  • Exam 2, Spring 2008Solutions
  • Exam 3, Spring 2008Solutions
  • Final Exam, Spring 2008Solutions
  • Exam 1, Fall 2008Solutions
  • Exam 2, Fall 2008Solutions
  • Exam 3, Fall 2008Solutions
  • Final Exam, Fall 2008Solutions
  • Exam 1, Spring 2009Solutions
  • Exam 2, Spring 2009Solutions
  • Exam 3, Spring 2009Solutions
  • Final Exam, Spring 2009Solutions
  • Exam 1, Fall 2009Solutions
  • Exam 2, Fall 2009Solutions
  • Exam 3, Fall 2009Solutions
  • Final Exam, Fall 2009Solutions
  • Exam 1, Spring 2010Solutions
  • Exam 2, Spring 2010Solutions
  • Exam 3, Spring 2010Solutions
  • Final Exam, Spring 2010Solutions
  • Exam 1, Fall 2010Solutions
  • Exam 2, Fall 2010Solutions
  • Exam 3, Fall 2010Solutions
  • Final Exam, Fall 2010Solutions
  • Exam 1, Spring 2011Solutions
  • Exam 2, Spring 2011Solutions
  • Exam 3, Spring 2011Solutions
  • Final Exam, Spring 2011Solutions
  • Exam 1, Fall 2011Solutions
  • Exam 2, Fall 2011Solutions
  • Exam 3, Fall 2011Solutions
  • Final Exam, Fall 2011Solutions
  • Exam 1, Spring 2012Solutions
  • Exam 2, Spring 2012Solutions
  • Exam 3, Spring 2012Solutions
  • Exam 1, Fall 2012Solutions
  • Other reading

  • About Wronski: MacTutor and Wikipedia
  • About Cramer
  • About Cramer's rule and Gaussian elimination
  • About population models.
  • Do you think forced spring models are complicated? Think again! They are linear. A forced pendulum is really complicated.