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Department of Mathematics
Mathematics 51
Spring 2015
Version 5.0
Sections: 01 (Block C, BP 2, Boghosian); 02 (Block C, BP 101, Nitecki); 03 (Block D, BP 101, Taylor);
04 (Block F, Anderson 312, Taylor)
Text: Reprint of M. M. Guterman, Z. H. Nitecki, Di↵erential Equations – A First Course, 3rd ed., Saunders
(1992).
Prerequisites: Calculus I – III (either ).
Course web page: http://courses.math.tufts.edu/math51/
Important dates: Two midterm exams: Thu. 1 Feb. 19, 12:00pm – 1:20pm and Mon., Mar. 30,
12:00pm – 1:20pm. Final examination: Mon., May 4, from 8:30am – 10:30am. Mark your calendar
now and please notify your instructor of any time conflicts with other examinations during the first week of
classes. Students who take two mathematics courses with any exams at the same time have to take these
back to back and must notify both instructors during the first week of classes. The scope of midterms is
indicated in the syllabus. Deadlines: Consult http://uss.tufts.edu/stuserv/AcadCal/default.asp.
Accommodation due to a documented disability: At the beginning of the semester please call
the Student Services Desk at 617-627-2000 to arrange an appointment with the Program Director of the
Student Accessibility Office.
Homework: Homework is due at the beginning of the next class. The homework that you turn in is
expected to be your work and yours alone. Helping your peers and seeking help from them is allowed;
mindlessly copying answers is highly unethical. Be certain that you are able to explain anything and
everything that you turn in, and that it is your intellectual property and yours alone.
Please mark your homework and folder 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 o↵ between instructor and
grader in a way that does not ensure confidentiality (e.g., by passing it out in class, or 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 (1) a bona fide attempt at every exercise (copying the
statement does not suffice) and (2) the correct solution to at least 60% of the exercises (answers only are
not enough). Do not claim credit for any parts of solutions copied from the blackboard during class! Your
homework credit is H = u26 (n) · (n 20)/4, where n is the number of homework points and u is as on p.
449 of the text.
Grades Your course average is computed from your lower midterm score L, your higher midterm score
T , the final exam score F , and your homework credit H as the larger of the following two numbers:
.10L + .40T + .50F + H and .33L + .33T + .34F + H. This average will be converted into a letter grade
according to the usual conversion scheme. If you must miss a midterm exam for a reason accepted by the
Mathematics Department, then we instead use the larger of .30T + .70F + H and .45T + .55F + H.
1
This day will observe a Monday schedule, according to the Academic Calendar.
1
Missing examinations: Please don’t, if you can at all help it, because it really creates difficulties for all
involved! Full information can be found on the mathematics web pages:
http://math.tufts.edu/courses/examPolicy.htm. If your ability to take an examination is in doubt for
any reason, please consult Gail Kaufmann <[email protected]>, 617-627-2162 and your instructor
as early as possible. Doing so in advance can help avoid one of 2 possible costly mistakes: Missing an
examination when the reason for doing so is not sufficient, or taking an examination under circumstances
in which you really should not. These can not be undone after the fact.
Cheating: There is much that could be said about this topic. To summarize: Don’t under any
circumstances! You must sign your exam book before handing it in. With your signature you are
pledging that you have neither given nor received assistance on the exam. If you question the grading on
a particular problem, write a note explaining the issue (on the cover of the exam if it will fit there), and
resubmit the exam booklet with the note at the end of the class when the graded booklets were returned.
Submitting an examination for regrading after altering original answers, submitting a fraudulent excuse for
course work missed, cheating on an examination or bringing unauthorized materials into the room during
an examination are all Level III o↵enses for which one can expect an F in the course and suspension for a
year or expulsion, and a permanent transcript notation. There is no appeal of grading consequences.
Learning objectives: Of our learning objectives found on the main department web page, this course
addresses mainly 1 a, b, e.
The Course: Sometimes laws of nature give us direct relations between the quantities we measure in
experiments. Ohm’s law, for example tells us that voltage is the product of current and resistance. Most
of the time, however, natural laws are not this simple. Instead they give us relations between the rates of
change of the quantities that we measure in experiments, and this leaves us with a mathematics problem
to solve.
For example, Newton’s law of gravitation describes how gravitational force depends on position. By
itself, however, this observation is useless. It becomes useful when you combine it with Newton’s second
law of motion which tells us that force is equal to the time rate of change of momentum, and with the
laws of kinematics which tell us that momentum is mass times the time rate of change of position. Taken
together, these expressions relate position with its own derivatives with respect to time. Such a relation
is called an ordinary di↵erential equation 2 for position, and only when one solves it does one obtain an
explicit expression for the position of a mass experiencing a gravitational force.
In similar fashion, di↵erential equations arise in many other contexts throughout the quantitative
sciences and engineering. They are used by astronomers when calculating the trajectory of a planet, by
physicists when computing the trajectory of a charged particle in an electromagnetic field, by population
biologists when studying the dynamics of predator-prey populations, by chemical engineers when studying
the variations of chemical concentrations in reactors, by electrical engineers when studying how a circuit
of resistors, capacitors and inductors behaves after a switch is thrown, and by mechanical engineers when
studying the flexure of a bridge under a varying load.
The course develops a complete method for solving constant-coefficient systems of di↵erential equations
and introduces qualitative methods for the study of nonlinear systems. The former entails much linear
algebra, which is in itself most useful; a course on Linear Algebra is an excellent follow-up to this one. The
qualitative analysis of nonlinear systems prepares participants for a fuller study of Nonlinear Dynamical
Systems and Chaos, and a course on this subject is another recommended sequel.
2
The word “ordinary” in this context means that only ordinary derivatives are involved in the expression, which usually
means that the unknown function depends only on one independent variable. If the unknown function depends on many
independent variables, then the law of nature might relate each of its partial derivatives, in which case we call the equation a
partial di↵erential equation. This course will deal exclusively with ordinary di↵erential equations.
2
Tufts&Mathematics,&MA&51,&Spring&2015&(v5,&2&April&2015)
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Block&C
01/14/15
01/16/15
01/20/15
01/23/15
01/30/15
02/03/15
02/04/15
02/06/15
02/11/15
02/13/15
02/17/15
Block&D
01/15/15
01/20/15
01/21/15
01/22/15
01/26/15
01/29/15
02/03/15
02/05/15
02/12/15
02/17/15
Block&F
Section
Homework
01/15/15
1.1,(1.2
p.(8:(1,(3,(7,(17,(21;(p.(16:(3,(5,(15,(17,(28
01/16/15
1.3
p.(25:(1,(3,(7,(8,(12,(19,(23,(25
01/20/15
1.4
p.(37:(1,3,9,10,13,15
01/22/15
1.6
p.(56:(1,3,7,11,17,19,21
01/23/15
1.7
p.(63:(126
01/29/15
2.A((((
p.(194:(1,(3,(5,(11,(17,(19,(31
01/30/15
2.2
p.(102:(1,(3,(7,(9,(13,(15,(17,(21,(23,(24
02/03/15
2.3
p.(113:(1,(3,(9,(15,(17,(31,(32
02/05/15
2.4
p.(120:(1,(5,(9,(10,(13
02/06/15
2.5
p.(129:(3,(5,(9,(11,(15,(18,(19
02/12/15
2.6
p.(136:(1,(5,(11,(13,(15,(19,(20,(23
02/13/15
2.7
p.(145:(123,(5,(11,(18
02/18/15
02/19/15
02/17/15
REVIEW
EXAM&ONE:&&THURSDAY&02/19/15,&12:00pmO1:20pm&(thru&2.7)
02/23/15
2.6
p.(136:(1,(5,(11,(13,(15,(19,(20,(23
02/20/15
02/24/15
2.7
p.(145:(123,(5,(11,(18
02/24/15
02/26/15
02/20/15
2.8
p.(155:(1,(5,(7,(9,(15,(17
02/25/15
03/02/15
02/24/15
2.9
p.(164:(124,(5a,(6,(8,(11
02/27/15
03/03/15
02/26/15
5.2
p.(421:(8210,(13215,(17,(19
03/03/15
03/05/15
02/27/15
5.2;(5.3
p.(421:(22,(23;(p.(432:(1,5,9,15
03/04/15
03/09/15
03/03/15
5.4
p.(441:(1,(5,(9,(15,(18,(27,(32
03/06/15
03/10/15
03/05/15
5.5
p.(452:(3,5,7,15,23,29;(p.(462:(23
03/10/15
03/12/15
03/06/15
5.6
p.(462:(7,(11,(13,(22,(23
03/11/15
03/23/15
03/10/15
3.2
p.(219:(1,(2cd,(5,(7,(8,(11,(13,(19,(24ab,(25
03/13/15
03/24/15
03/12/15
3.3
p.(232:(1,(7,(9,(11,(13
03/24/15
03/24/15
03/13/15
3.4
p.(241:(125,(7,(8,(11214
03/25/15
03/26/15
03/24/15
3.5
p.(254:(9,(10,(11,(13;(p.(265:(14,(17,(18,(19
03/26/15
3.6
p.(265:(1,(2,(3
03/27/15
03/30/15
03/27/15
REVIEW
EXAM&TWO:&&MONDAY,&03/30/15,&12:00pmO1:20pm&(thru&3.5)
3.5
p.(254:(9,(10,(11,(13;(p.(265:(14,(17,(18,(19
03/31/15
03/31/15
3.6
p.(265:(1,(2,(3
04/01/15
04/02/15
03/31/15
3.7
p.(275:(1,(4,(5,(729,(11,(13
04/03/15
04/06/15
04/02/15
3.8
p.(103:(27;(p.(284:(1,(3,(5,(7,(11
04/07/15
04/07/15
04/03/15
3.9
p.(295:(1,(5,(7,(9,(13
04/08/15
04/09/15
04/07/15
3.11
p.(320:(428,(13;(Recommended:(p.(321:(1214
04/10/15
04/13/15
04/09/15
4.1;(4.2
p.(332:(1,5;(p.(348:(123
04/14/15
04/14/15
04/10/15
4.2
p.(348:(729,(11,(13,(15,(20
04/15/15
04/16/15
04/14/15
1.8
p.(71:(1,(7,(13,(14,(19
04/17/15
04/21/15
04/16/15
4.3
p.(364:(9,(10,(14,(16,(17
04/21/15
04/23/15
04/17/15
4.6
p.(397:(1,(3,(4,(5,(8
04/22/15
04/27/15
04/24/15
04/21/15
04/23/15
REVIEW(
Chapters(1,(2
REVIEW(
Chapters(3,(5
p.(84:(1,(13,(22;(p.(183:(1,(2,(6,(15
p.(321:(124;(p.(462:(18,(19,(29;(p.(473:(13,(18
04/24/15
FINAL&EXAM:&&MONDAY,&05/04/15,&8:30amO10:30am&(Cumulative)