Homework assignments, Differential Equations (Math 256) Fall 2013
NOTICE: There is a yellow bar
somewhere below, which should cut this calendar in two somewhere around
the current date. Above the yellow bar are actual
assignments. Below the yellow bar are tentative
assignments. While the tentative items will mostly come true,
there will be some changes.
Pay particular attention to the direction fields or slope fields in exaples 2 and 3 of
Also think hard about the method of example 2 in section 1.2.
Prepare for a syllabus quiz on Friday.
Do 1.1 #1, 11,15-20.
For each of 1.1 #21,23,24:
Identify the quantities that are changing in the problem, and
what information the problem gives about their rates of change
Use this information to write the requested differential
equation, and complete the problem.
Do 1.2 #3a, following the example 2 in the section. Be
careful, it's a bit different - so find a way to check your work.
do the orientation for WebWork by shortly before midnight on
Thursday. You should have an email from me with step-by-step
Aug. 30: Syllabus Quiz;
Read 1.3 (1.4 optional).
Look at Prof. Janeba's
to get important advice about reading a text. This survival guide
is written for calculus students, but it all applies to DE.
Do 1.3 #1-5,7,9,11,12,13,15,19 (practice showing work for your quiz next
Friday; hint for 15,19: treat r
as a constant, try to verify the specified solution, and note that this
will only work for certain
values of r. What are
Read 2.1 and do #13,15,19
Start Integral Review 1 assignment on WebWork - due late next Wednesday, but you'll want to start now
NOTE: Exactly ONE of the problems in this WebWork is
"too hard" - If you get all but that one, your grade will be considered
100% for this assignment (no matter what WebWork says)
The goal here is for you to learn to identify when the elementary methods of integration don't apply, rather than blindly using them despite the fact.
Sept. 2: Labor Day - no classes
Do 2.1 #14,17,20,21; Read 2.2 and do 2.2 #1,3,8
There is a second WebWork integral review due late next Sunday night.
Sept. 6: Quiz #1;
#4,9,11,21, read 2.3 & ponder problem 2.3#1
Do 2.3 #7,8 Hint: Follow example 2. Be sure to read what compounding continuously means;#7 can be done
w/o differential equations, because the solution to the general problem is
already given in the example, but #8 requires a new D.E. - follow the
example to build it. Note also that part 8c will require you to
solve an equation computationally/approximately - an exact algebric solution is not possible.)
Do the MOAMP (Mother of all mixture problems):
Prof. Janeba's office initially contains 3000 gallons of Diet
Pepsi (384,000 fluid ounces). Diet Pepsi contains 35 mg of
caffeine per 12 ounces. Professors Johnson and Laison pour in
coffee at a rate of 1200 fluid ounces per day; this coffee contains 100
mg per 8 ounces. An unknown student lobs in (kind of like hand
grenades) No-Doz at a rate of 30 pills per day; No-Doz contains 200 mg
of caffeine per pill.
Suppose, first, that all this fluid leaks out the window at
the same rate as it is being added (in ounces per day). You may
assume the volume of the No-Doz is negligible, (though its caffeine
content is not!) i.e. the No-Doz does not contribute to the rate
leakage. You may also assume that the caffeine is "uniformly
distributed" in the fluid in Prof. Janeba's office. Then give an
initial value problem for the amount of caffeine in Prof. Janeba's
office as a function of time, AND give the amount of caffeine present
in his office after 75 days.
Extra Credit (optional,
turn in neatly written by Friday's class if you desire) Modify the
MOAMP into the GMOAMP: Assume that the fluid is leaking out at a rate
of 3000 fluid ounces per day (inputs are unchanged from the
MOAMP). We still keep the "uniformly distributed" assumption. Then
Give an initial value problem for the amount of caffeine in
Prof. Janeba's office as a function of time (beware, the volume of
fluid in the office is not constant), AND
give the amount of caffeine present in his office after t days.
For a super-bonus, use your answer to the previous part to tell what the concentration
of caffeine is in the office just as the volume of fluid it contains
reaches zero. There is an intuitive answer here, but to get the
super-bonus, you have to show how the previous answer shows it as well.
Do 2.3 #23 Note this is a very complicated problem,
though very practical. Solve it in stages. Note the D.E.
changes when the parachute opens, so you'll need to solve separately
Read 2.4. This will be rough sledding, but focus
particularly on the examples - not only on the work done, but the
conclusions drawn about the nature of DE solutions.
Do 2.4 #1,3-5,8,10, 13,15
Sept. 13: Quiz 2 postponed to Monday
Read 2.5 (this one is fun; we can get lots of qualitative answers with relatively
little work, for a certain kind
of diff. eqn.)
Do 2.5 #1,3,5, and 7 (7 is especially fun).
Sept. 16: Quiz #2;
Some routine homework: 2.5 #8-12
2.5 #14 This problem requires some thought; be careful
distinguishing between the graph of f(y)in the phase plane and the graph of y(t).
So how can semistable equilibria occur in the real world?
Do 2.5 #21, an application to fish farming, AND explain in a sentence
or two what each of the h
values in (a),(d), and (e) mean in practical terms to the fish farmer.
Start reading 2.6
Do 2.5 #22,23 (use Maple, Wolfram Alpha, or the like for
the integral in 22b) Note #23 is a somewhat complicated two-step
Do 2.6 #1-4
Do 2.6 #5-7,10-14,18 (optional: 19,20: which show how integrating factors can help, but doesn't show how to find them.)
Read 2.7 (The basics of Euler's method will be on next week's exam, you'll want to read up on those basics.
Try 2.7 #1,2 parts a,d only. Part (b) is easy to do with a spreadsheet or a graphing calculator, if you use it carefully.
Sept. 23: Do 2.7 #1,2,11,17 You may skip the parts about direction fields, but do use
technology of some sort to compute the Euler estimates.
Review exercises and study advice: Work on review problems, p. 132-133:
a possible solution method (e.g. linear (integrating factor ), separation, exact)
Note the authors have made it tricky in that you will often need to do
some factoring or other rearrangement to "see" what method applies.
For example, you will need to factor #13 before you'll get
Now go back and solve
several of #1-24, making sure to do several of each type (integrating
factor, separation, exact). Don't forget to include several that
have initial conditions.
can keep going to #32 if you like, I have checked that #1-24 can all be
solved with our methods, although the integrals may be hard.
Be prepared to write a long paragraph to explain how/why each
of these methods work: linear/Integrating factor, separation of
variables, exact d.e.s, as well as the setup of Euler's method.
We have discussed all of these in class, except for separation (the
easiest), which is covered well on the top half of p.44.
Don't forget to review modeling with D.E.'s (chapter 2.3)
Refer to the survival
guide for overall study advice - 70% doing problems, 30% organizing the material / deciding what you need to study.
Sept. 25: Exam 1 (full hour)
Use error extrapolation as covered in class to refine your
Euler estimates in any three of the homework problems from 9/23
above. Note that our treatment assumes that one cuts the stepsize
in half from one estimate to
the next, so watch out on 11d, where h
went from .025 to .01; similarly for the last estimate on
Read 3.1, and do 3.1 #1-5,9-12
Do 3.1 #14,15,17 (and the 3.1 homework above that you haven't already done).
3.2. There's a lot here, try to form an overview, and not get
overwhelmed in the details.
Re-read (really) 3.2 in light of today's lecture, AND
read 3.4 (yes, we are jumping around a bit)
Do 3.2 #1,3,5,7-9,15,21,24
Notes: #5 and #24 are important examples,
Be sure to read the directions on 7-9 carefully
Do 3.4 #1,2,3,8,11,12,15
Oct. 4: Quiz #4; Read 3.3 & 3.5 and do 3.3 #1-3,7-9,11,13,17-19
do 3.5 #1,2,3 (Hint for #3: Try yp = (At+B)e-t ),
#4 (Hint for #4: you need to get out the sum of a constant and a trig
function; use something for each in your guess for yp.)
do 3.5 #5,7,9,15,16,18, and read 3.6
Extra Credit: Do 3.5 #30 with
this change: y''+2y'+5y = 5 when 0t/2[and
keep y''+2y'+5y = 0 for t > /2 ]
write up an explanation of
your solution that a fellow student could understand, and turn in by
Tuesday afternoon for up to 8 points.
Oct. 11: Quiz #5; Read 3.6 (again!) and do 3.6 #1,5 Hint for
#5: Use an integral table (online or in a book) to find sec(t)dt.
do 3.6 #1,5 if you haven't already, and 3.6 #7,10,13,15,16
Note: For #13,15,16, the text method only applies to d.e.'s in the form y''+p(xy'+q(x)y = g(x); the de's in these problems are not yet in this form.
Oct. 16: Read 3.7; do 3.7 #1-3, 5-8,13,17,18,24.
Oct. 18: Mid-semester day (no classes)
Study for exam. Spend some time on the various "how
things work" explanations - can you produce them on a blank sheet of
Do the 3.7 homework above. Don't stress overmuch (stress some) if you have trouble sorting the coefficients k and
from the physical description, but do be able to otherwise set up and
solve the differential equation. It won't hurt to try the one
electrical problem by following the text example. It would be a
good idea to have some idea how the three categories of underdamped, overdamped, and critically damped
are related to our three categories of characteristic equations
(distinct real roots, repeated real roots, complex roots).
do 3.8 #17,18, which illustrate the difference between damped
and undamped oscillators, as we approach the resonant frequency.
Note that oscillators with very low damping will behave approximately
as undamped oscillators, in the short run at least, so the lack of
"perfectly undamped" oscillators in nature doesn't make the undamped
Read 4.1 & 4.2. Note that virtually all our methods
from chapter 3 transfer immediately to higher-order linear d.e.s with
constant coefficients, so these are essentially review sections.
On your final exam next week, you may bring one 8.5"×11" sheet of paper, handwritten
on one side with whatever you wish to write. Do not bring
computer-printed sheets, do not bring photocopied sheets, the sheet
must be handwritten.
Spend, as always, 70%
of your study time actually doing problems. Spend the other 30%
organizing your thoughts about the categories of problems we have
solved (and other theoretical questions we have studied this term), and
the tools we have learned to solve them (and the theory we've
learned). Part of that 30% will be writing your sheet of
paper. If you expend some serious thought about what should go on
the sheet, and how to organize it, (rather than writing everything you
can find in really tiny print), it will be study time well-spent.
A Laplace transform table will be provided on the final exam, the same table you had for exam 3.