MAT2705 20F homework and daily class log
Jump to current date!
[where @ is
located]
Daily lectures and your homework will appear here each day as it is assigned, including some PDF
and/or Maple problem solution files or PDF notes, with occasional links to
some MAPLE worksheets when helpful to illustrate some points where technology can be
useful. [There are 56 class days in the semester, 4
each normal week, numbered consecutively below and labeled
by the (first initial of the) day of the week.
Friday will usually be the quiz day. Monday quiz makeup day.]
It is your responsibility to check homework here since not all homework
exercises are online. (Put a favorite in your
browser to the class homepage.) You are responsible for any hyperlinked
material here as well as requesting any handouts or returned tests or quizzes from classes
you missed. Homework is understood to be done by the
next class meeting (unless that class is a test, in which case the
homework is due the following class meeting), but the online deadline will be
stated as midnight of the due day so that you have time in class or after to ask
questions that you did not submit via the Ask the Instructor
online tool. You may correct submitted homeworailyk after the deadline without
having to request an extension.
[[Homework
problems surrounded by double square brackets
[[...]] are not in the online homework
system but are important to do.]]
Read:
HOMEWORK ADVICE;
ZOOM CLASSROOM/OFFICE HOUR INFO
quiz, test BlackBoard access/submission

M: (August 17)
DURING CLASS:
Lecture
Notes 1.1a: Differential Equations: how to state them and "check" a solution;
summary handout: odecheck.pdf
New to Maple 3 minute video
AFTER CLASS (THIS IS THE HOMEWORK):
1) Log on to My Nova, choose the Class Schedule with Photos,
view fellow students.
2) Go to
BlackBoard and look at the class portal and Grade book
for our course: you
will find all your Quiz, Test and Maple grades here during the semester once
there is something to post. Everything we do except for quizzes, tests and
grades will take place through our course website.
3)
Download Maple
2020
for Windows or Mac
if you haven't already done so and install it on your laptop when you get a
chance (it takes about 15 minutes). If
you have any trouble, email me with an explanation of the errors.
You are expected to be able to use Maple on your laptop when
needed. We will develop the experience as we go. No
previous experience is assumed.
4) Enter the etext MyLab Math portal
if you have not already done so!
registration codes:
004:
jantzen48548,
005:
jantzen35255
[click codes for detailed instructions]
5)
Homework Problems:
1.1: 3, 5,
13, 33 online.
[memorize!: "A is proportional to B" means "A = k
B" where k is some constant,
independent of A and B]
(short list so you can check out our class website and read about the
course rules, advice, bob FAQ, etc, respond with your email).
It is
important that you read the section in the book from which homework problems
have been selected before attempting them.
Optional: get acquainted with
Maple clickable calculus DE entry and "odetest" for problems 3,5,13 [even
33!]
Proportionality statements
must be converted to equalities with a
constant of proportionality introduced:
y is
proportional to x: y ∝ x means y
= k x . [y is a multiple of x]
y is
inversely proportional to x: y ∝ 1/x
means y = k/x .
y is
inversely proportional to the square of x: y ∝ 1/x^{2}
means y = k/x^{2}
6) By the end of the week, reply to my
welcome email from your OFFICIAL
Villanova email account (which identifies you with your full name),
telling about your last math courses, your comfort level with graphing
calculators and computers and math itself, [for sophomores only] how much
experience you have with Maple (and Mathcad if appropriate) so far, why you chose your major, etc,
anything you want to let me know about yourself that will give me more of an
idea about you as a person. [For example, I like to do
humorous sketching.
and cooking.] Tell me
what your previous math course was named (if at VU: Mat1500 = Calc 1, Mat1505 = Calc
2, Mat2500 = Calc3).
[In ALL email to me, include the string "mat2705" somewhere in the
subject heading if you want me to read it quickly. I filter my email.]
 T:
Lecture
Notes 1.1b: Differential Equations: initial conditions
1:1: 8, 23 [see
Maple plot (execute worksheet by clicking on the !!! icon on the
toolbar)];
formulating DEs: 27 [approach like 29, make a generic diagram like
this one to calculat the slope of the
tangent line and set it equal to the derivative
.mw],
[[29. Hint: recall perp lines have slopes which are negative
reciprocals, the normal line is perpendicular to the tangent line and passes
through the same point on the curve; make a diagram of the given point (0,1), a "generic" function graph curve
whose normal at a point (x,y) on the curve passes through
(0,1), and draw in the connecting normal line segment between these two
points and the perpendicular tangent
line, then compute the slope of the normal line from the two points, and then from the
negative reciprocal of the derivative value, finally equate the two
to get the DE: mw ]],
31, 35, 43;
online handout:
initial data: what's the deal?
Is your dorm abbreviation missing from
bob's dorm list?

W: read the handout: algebra/calc background
sheet [online only: more rules of
algebra NOT!];
Lecture
Notes 1.2: First order DEs independent of the unknown [some
redundancy with previous lecture, plus two word problems];
[lunar
landing example: mw] [Swimmer
problem example: mw]
1.2 (antiderivatives as DEs): 1, 5, 15,
21 [Hint: write piecewise linear function from graph: v_{1}(t) for first expression,
v_{2}(t) for second expression, solve first IVP
for x_{1}(0) = 0, second IVP for x_{2}(4) = x_{1}(4);
example: pdf,
mw],
25 (like lunar landing problem,
see example), 35,
41 (mw: short,
long)
[[43 (pdf)
[convert to appropriate units!]].
Nongraded Blackboard Quiz 0
due today by 11:59pm. Does anyone not have access to a printer?
remind bob to demonstrate quiz upload

F: Quiz 1 (read
Test/Quiz rules;
submission instructions): due Sat 11:59pm,
start in class;
if time, more discussion of 1.2
river crossing problem;
Lecture
Notes 1.3: First order DEs; Direction (= slope) fields and complications
[directionfields.mw] [ode1complications.mw]
1.3 direction
fields: 3, 11, 15;
[[optional: 1.3:
3:
for fun hand draw in all the curves with a pencil on the full page paper
printout:
projectable image;
bob's attempt]
try
the Maple template for 5]]
WEEK 2:

M: Quiz one
answer key;
Lecture Notes 1.4a: Separable first order DEs
[
example
1: implicit soln, other
complications visualized].
1.4 (separable DEs): 1, 4,
9 (see mw!),
25, 27, 29.
Office hours; Course "Syllabus" now
online; Final Exam dates
HW "due date" = Dec 6 to
allow resubmission of answers until you get it right
but HW "due" the
day after it is assigned to keep on track.

T:
Lecture Notes 1.4b: Separable first order DEs
handout on
exponential behavior/ characteristic time
[how to
plot?] [cooking
roast in oven remarks]
[read this worksheet
explicit plot example, execute
worksheet first with !!! icon on toolbar];
1.3.3 online;
1.4: 33,
35, 41,
45 (if your cell phone were waterproof: "Can you hear me now?"
attentuation of signalcharacteristic length),
49.

W: 1.4: 69,
[[65.
CSI problem (Newton's law of cooling) same as worked problem but at
end we want to find a previous time instead of a subsequent time]].
Maple play to help familiarize you with using it. Split screen between my
projected Maple and your local Maple so you can try it out.
We take a
breather to catch up.
Hyperbolic functions are important in DEs! [wiki]
Optional: after you solve this
problem yourself, consult both [pdf,
.mw];
please read Maple worksheet to see
the power of mathematics to solve all such problems at once;
please read the PDF to see how to convert the word problem line by
line to a mathematical problem].
If curious: revisit the
river crossing problem and the
oven heating problem.
Bonus discussion: average
value of velocity profile:
river crossing problem
[please read this Maple worksheet to see the power of mathematics to solve
all such problems at once]

F: Quiz 2 (separable DEs, see Quiz 2 in archive);
Lecture Notes 1.5a: Linear first order DEs
online handout: recipe for first order linear DE [plot]
[mw exploration]
1.5: 2, 5, 8, 21, 24;
*[check at least one of these problems, both the general solution and the initial
value problem solution with the dsolve
template].
For future reference:
> deq := y ' = x y
[space implies multiplication]
> sol:=dsolve(deq, y(x))
> solinit := dsolve({deq, y(0)=1}, y(x))
[or enter DE plus IC separated by a comma, right click on output,
choose Solve DE, for
y(x), or Solve DE interactively;
rightclick "Simplify, Simplify"
or "Simplify, Symbolic"
(with radicals) may be necessary to simplify the result]
> y ' = x y , y(0)=1
Use function notation to change independent variable:
> y '(t) = t y(t) , y(0)
= 1
class directory list (image only by email for
privacy reasons);
WEEK 3:

M:
Lecture Notes 1.5b: Linear first order DEs:complications
[acc. funcs.: mw]
[Khan]
[slope field: mw]
1.5: 15, 24, 27; 30;
[[29:
solutions defined by a definite integral. Use the fact that the derivative of
the function defined by the integral is the integrand including all
multiplicative factors
]];
[[41
in class activity with breakout rooms if bob can make
them work: epc41541.htm
]]

T:
Lecture Notes 1.5c: Linear first order DEs:
Mixing Problems [Maple
template]
[this mixing tank problem is an example of developing and solving a differential equation
that models a physical situation
(online handout); ];
1.5 (ChE majors especially) Use
Eq. 18 in the book or the boxed equation in the online handout:
respectively constant, decreasing, increasing volumes: 33, 36, 37;
constant volume lake application: 45; [solution
worksheets];
we attempt another breakout
session on problem 33 collaboration;
optional extra questions
interesting to answer: what is the final concentration?
How does it
compare to the initial concentration and the incoming concentration?
Note: we are skipping 1.6: exact DE
etc. These are less important, and Maple can solve
these when needed anyway. A similar integrating factor technique works for
exact equations but
where the independent and dependent variables are on an equal footing.

W:
1.R (review):
be prepared in class to classify the odd problems 135 as:
separable, linear in y
(as unknown),
linear in x (as unknown), some combination of these three, or NOTA (none of the above,
i.e., from skipped section 1.6), for
example dy/dx = y/x is all three and can be solved in three different ways;
don't solve them but (review problems are not online):
[[ solve 25 [it can be done in two ways by expanding out the square on the RHS
before integrating with integration constant C or by using a usubstitution
as the book did with integration constant K; express K in terms of C as given in the book supplied answers by
combining the two terms in the K solution [identity for expanding out (x+1)^{3} !] and then comparing with the C
solution];
solve 35 in two ways and compare the resultsdifferent
constants enter your expressions, show how they agree]];
for another word problem example,
see this (both linear and separable;
similar to late stages of logistic behavior next section)

F:
Quiz 3 (linear DE see quiz 3 in archive); check your graded Quiz 2 in
BlackBoard;
Lecture Notes
2.1a: First order DEs: Logistic Equation
handout on solution of
logistic DEQ
[Maple: characteristic time
/shape, directionfield,
integral formula,
24];
2.1 (logistic DE):
5 (resolve the DE this one time following the steps in the Lecture Notes),
6
(reverse S curve, choose a horizontal window to show full S curve based on
plus/minus 5 times characteristic time,
answer: >
dsolve({x'(t) =3
x(t) (x(t)5),
x(0)=2},x(t))
(copy and paste into the input line, rightclick choose RHS, then
from the CSM
choose Plotbuilder, choose horizontal window based on characteristic
time, make sure you then see the Scurve without wasting too much window
where nothing is happening);
17 (see Maple worksheet for setup
problem 15);
21 (initial value of population and its derivative given
instead of population at two times, need to set k and P(0)
with two conditions),
23 [Note dx/dt = 0.004
x (200x) so k = 0.004,
M=200 for logistic curve
solution formula],
29.
WEEK 4[1]:
M: Labor Day

T:
Lecture Notes 2.1b: First order DEs: More models, Separable: f(y)
2.1 (other population models):
handout on DE's that don't involve the ind
var explicitly;
11 ["inversely propto sqrt": use β = k_{1}/P^(1/2), δ =
k_{2}/P^(1/2) in eq.(1), these are fractional
birth/death rates;
leads to P' ~ P^(1/2)
soln],
13 [P' ~ P^2 example],
[[30, 31 [ans: find β_{o} = 0.3 from condition dP/dt(0) = 3x10^5
using the DE at t =0, then use solution
α=0.3915 from second condition (solve this yourself or at least check numerically that it satisfies the 6
month condition) to find the limiting population as t goes to infinity (i.e.,
neglect decaying exponential)]].
39 [oscillating growth].

W:
Lecture Notes 2.3: First order DEs: acceleration models
2.3 (accelerationvelocity models: air resistance):
air resistance handout
(example of a piecewise defined DE and solution and the importance of dimensionless
variables);
[optional reading to show what is possible:
comparison
of linear, quadratic cases; numerical solution
for any power];
2;3: 1, 3,
9 [remember weight is mg, so mg = 32000 lb determines
mass
m = 1000 in USA "slug" units, convert final speed to mph
for interpretation!],
17 [v > 0 falling slowing down
to stop, plug in numbers],
19 [constant thrust = reverse gravity!];
[[optional reading:
22 this is the tanh case: v_{terminal} = 20.7 ft/s, t
= 8 min 5 s].[soln,
.mw]]]

F: Quiz 4 on population modeling, like HW problems;
Lecture Notes 2.4: First order DEs: numerical DE solving (Euler)
2.4 (Euler numerical solution):
1, 27 [this shows the Maple approach],
29 [naive
approach]
[[optional reading:
example. problem 1 studied in detail
showing the algorithm with eulerdoc.mw,
soln)]]
WEEK 5[1]:
detour into linear algebra; Test 1 on first order DEs next weekend, starting
in class Friday?

M:
Lecture Notes 3.1: linear systems of equations: elimination, DEs;
[inconsistent 3x3 system; DEs
lead to linear systems];
why LinAlg with DE?;
[maple dsolve and DEplot for 2x2 systems
of DEQs];
Read 3.1 on linear systems (review of high school topic solving 2 or 3 linear equations in
2 or 3 variables),
do
3.1: 3, 5, 7; 9, 23, 25.
Check answer key
for Quiz 4.

T: 3.2:
Lecture Notes 3.2: matrices and row reduction "elmination";
handout on
RREF (Reduced Row Echelon Form, section 3.3)
[see bob's examples 2,
3 using Maple];
3.2: 1, 5, 7, 9; 11,13,15,
(1, 5, 7 already Gaussian reduced, for rest do a few
RREF by hand, then you may use stepbystep row ops with MAPLE or a calculator for
the rest) ;
[Matrices for doing some of tonight's homework with Maple
preloaded];
you must learn a technology method since this is insane to do
by hand after the first few simple examples);
23 [can your calculator handle this?]

W: 3.3:
Lecture Notes 3.3: Row reduction solution of linear systems;
handout on
solving linear systems example
[.mw];
we always want to do a full [GaussJordan] reduction, not the partial Gauss
elimination reduction also described by the textbook;
the Matrix
palette inserts a matrix of a given number of rows and columns;tab between
entries;
in breakout rooms of 2 open this Maple file
and use the LinearSolveTutor and enter the matrix for the system 3.2.15
given there and reduce step by step,
then solve the system (if possible), then switch positions and enter the matrix
for the system 3.2.18 given there and repeat using instead the Reduction
command template there;
3.3: 1, 3, 9, 11, 14, 16, 17, 19, 23 (= 3.2.13), 29 (=3.2.19)
Since the homework portal changes the matrices, I cannot preload them for
you. But you can use this template to invoke the Tutors and
the reduction and solving commands.

F: Test 1 on 1st order DEs in usual Quiz format due Sunday midnight:
1 population modeling problem, 1 linear DE, 1 separable DE, so 2.5 days
to do three homework problems well.
Read
Test Instructions;
Remember bob is trusting you to observe
our Villanova honor system.
Lecture Notes 3.3: Balancing Chemical Reaction Application (short
lecture);
In class breakout room fun: Application:
chemical reaction problem.
(what can you
find out on the web about interpreting this chemical reaction?
chem site reaction balancer
gives balanced reactions but
no description, this one of our exercise turns out to be interesting).
word of the day (semester really): can you say "homogeneous"?
[online handout only]
Get used to this word, it will be used the rest of
the semester. It refers to any linear equation not containing any terms
which do not have either the unknowns or any of their derivatives present,
i.e., if in standard form with all the terms involving only the unknowns on
the left hand side of the linear equation, the right hand side is ZERO. [nonhomogeneous
means the RHS is nonzero]
WEEK 6[1]:

M:
Lecture Notes 3.4: Matrix operations [finally matrix multiplication: handout];
in progress:
3.4: 1, 5, 7, 9, 13, 15, 17, 19;
[[read:
27, check with Maple: 43]];
Matrix algebra is easy in Maple
[see here for how to do Matrix stuff in Maple].

T: Lecture Notes 3.5: Matrix inverses; [Maple
inverse template: mw]; [2x2
inverse! mw,
pdf]
3.5: 1,
5, 7, 9, use Maple to do row reductions:
11, 19,
23 [Maple template; this is
a way of solving 3 linear systems with same coefficient matrix
simultaneously, as in the alternative
derivation of the matrix inverse];
30.
Matrix multiplication and matrix inverse, determinant, transpose [or
rightclick and use Standard Operations menu]
> A B [note space between
symbols to imply multiplication]
> A^{1
} > A [absolute value sign
gives determinant, or just right click]
Memorize:
.
Switch diagonal entries. Change sign offdiagonal entries. Divide by
determinant.
Always check inverse in Maple if you are not good at remembering this.

W: 3.6:
Lecture Notes 3.6: Matrix determinants;
[forget cofactors,
forget Cramer's rule (except remember what it is when oldfashioned texts
refer to it), use Maple instead];
optional online handout:
determinants and area etc ;
[optional: why
the transpose?
(to combine vectors we arrange them into rows of matrix,
the transpose maps columns to rows, rows back to columns)];
Read
explanation of why we need
determinants;
3.6: (determinants abbreviated:
forget
about minors, cofactors, only need row reduction evaluation to understand)
7, 9, 10, 13, 15, 17, 25, 29.
[[7*use
this example to record your Gaussian elimination
steps in reducing this to triangular form and check det value against your
result]].
Optional.
Explanation of why determinants are important for measuring area and volume
etc in higher dimensions. [Calc 3 stuff]

F: Quiz 5 row reduction and solving a linear system;
Lecture Notes
4.1: Linear independence;
R^n spaces and linear
independence of a set of vectors [motivating
example];
now we look at linear system coefficient matrices
A as collections of columns A = < C_{1} ... C_{n} >,
matrix multiplication on the right by a column of
coefficients as taking linear combinations of those columns,
A x = x_{1 } C_{1} + ... + x_{n } C_{n}
and solving homogeneous linear systems of equations A x = 0 as
looking for linear relationships among those columns (nonzero solns=linear
rels) ;
4.1: 5, 7; 13; 15, 17; 19, 23; 29, 31, 33
[use technology in
both 1) evaluating the determinant and 2) contexyt sensitive
menu for the ReducedRowEchelonForm
needed
for these problems];
hand out on the interpretation of solving linear
homogeneous systems of equations: A x = 0
[optional
read:
visualize
the vectors from the handout].
WEEK 7[1]:

M:
Lecture Notes
4.2: Vector spaces and subspaces;
study the handout on solving linear systems revisited
[remember the original one: solving linear systems example];
[note only a set of vectors x resulting from the solution of a linear homogeneous condition
A x = 0 can be a subspace of a vector
space; interpretation: lines, planes, ..., hyperplanes through the origin];
4.2: 1, 3, 7, 11; 15, 21.

T:
Lecture Notes 4.3: Span of a set of vectors;
so far: linear
independence (A x = 0),
express a vector as a linear combination (A x =
b)
[or does b belong to a linear relationship with
the columns of A?];
4.3 (span of a
set of vectors): 1, 3, 5, 7; 9, 13; 17, 21.
Test 1 answer key is
online; read
test120fremarks.htm.
For those curious about what the
value of a determinant means read
this.

W: Lecture Notes 4.4:
Bases;
handout on
linear combinations, forwards and
backwards [maple to visualize]
4.4: 1, 3, 5, 7;
9 [this is already reduced, find solution as in class example, pull apart to find coefficient vectors of
parameters, repeat for next 4 problems reducing first],
13; 15, 17, 25 [use technology for all HW row reductions and determinants
(but 2x2 dets are easy by hand!)].

F: Quiz 6 on solving linear systems and linear independence;
handout on nonstandard
coordinates
on R^{2} and R^{3} [goal: understand the jargon and
how things work];
0) bob tries to draw this out on the white board,
explaining how it works.
Then in the breakout rooms, you do the
following each separately, but discussing and critiquing each other's
work (hold it up to the screen, enlarged for the person trying to share
their work in speaker view!).
1) Using the completed first page
coordinate handout made for that new basis {[2,1],[1,3]} of the
plane just explained by bob, graphically find the new coordinates of the
point [4,7] reading off the grid. Then find the old coordinates for the
point whose new coordinates are [2,2]. Your graphical read outs can be
confirmed by the corresponding matrix multiplications if there is
disagreement.
2) Next practice on the blank grid [
.mw,
.pdf] you printed out in advance with b_{1}=
<3,1>, b_{2}= <2,2> . Draw these
arrows at the origin, and then replicate then tip to tail along the new
coordinate axes, marking off the new tickmarks and reproduce the grid
corresponding to 2..2 in the new coordinate tickmarks. What are the
new coords of the point (position vector!) <2,6>? What point (position
vector!) has new coords <2,1>?] .
3) Next we change to new basis
vectors and using the above handout as a guide, complete the blank
worksheet printed out in advance, similarly drawing its new basis
vectors and axes with tickmarks and the grid, etc. Use a ruler
and sharp pencil,to make a new 2..2,2..2 coordinate grid
associated with the new basis vectors {[1,1],[2,1]} and graphically
represent the point [x_{1},x_{2}] = [5,1]
and find its new coordinates [y_{1},y_{2}]
using the grid. Similarly read off the old coordinates [x_{1},x_{2}]
of the point whose new coordinates are [y_{1},y_{2}]
= [2,1]. Confirm that these agree with the corresponding matrix
calculations. Put your name on it, scan it to
lastnamefirstnamequiznot.pdf and email it to me by next class with
the filename in the subject header next class:
Subject: mat2705:
LastNameFirstNameQuiznot.pdf. [not for credit]
solution:
completed worksheet [Maple version
of the example]
not Fall
break week follows week 7!
not this year!
WEEK 8[1]:

M: 4.7 Lecture Notes 4.7:
Vector spaces of functions;
bridge to vector spaces
whose elements are functions.
handout summarizing
linear vocabulary for sets of vectors;
[optional online handout:
linear system vocabulary
for linear systems of equations ]
Part 2 of course begins: transition back to DEs:
read the 4.7 subsection (p.279)
on function spaces and then examples 3, 6, 7, 8;
if you are curious, example 5 illustrates the partial fraction decomposition
needed in engineering integration;
read the worksheet on the
vector space of (at most) quadratic functions [quadratics.mw,
pdf];
4.7: 9. 11, 14 ( c_{1} f_{1}(x)+ c_{2 }f_{2}(x)
= 0, set x = 0 to get one condition on the two coefficients,
set x = 1 to get a second, then solve 2x2 system),
15 [Hint: solve c_{1}(1 + x) + c_{2}(1  x) + c_{3}(1  x^{2}) = 0, for unknowns c_{1},c_{2},c_{3};
are there nonzero solutions? if not, these are linearly independent polynomials, in terms of which any quadratic
expression in x can be written; note that any two (nonzero!)
functions of x that are not proportional are automatically linearly
independent],
18 [same approach].
Reminder:
vectors are identified with column matrices!
Optional Reading: Why do we care
about linear coordinate grids in the plane?

T: Lecture Notes
5.1a:
2nd order linear DEs: an intro;
read 5.1 up to the subsection on
linear independence;
Memorize: y ' = k y <  > y = C e^{ k x}
y '' + ω^{2} y = 0 <  > y =
C_{1 }cos(ω x) + C_{2} sin(ω x)
["omega" = ω is the angular frequency "radians per time unit" as opposed to
just "frequency" f =
ω/(2π ) or "revolutions or cycles per time unit" as in physics, but in our class we
will just say "frequency" for ω, assumed to be expressed in radians per time
unit or converted to revolutions per time unit as convenient;
see also
damped
harmonic oscillators and
RLC circuits, and
Hertz, not just a rental
car company but "cycles per second"]
5.1:
1, 3, 9, 11, 17.

W: Lecture Notes
5.1b:
constant coefficient 2nd order linear homogeneous DEs;
handout on sinusoidal example;
repeated root plot;
5.1[from
linearly independent subsection]: 33, 35, 39,
[[49 find the IVP solution for y, set y ' = 0 and solve exactly
for x, backsubstitute into y and simplify using rules of
exponents
check your IVP solution for this curve in Figure 5.1.6; does the back of
book answer y = 16/7 in decimal approximation agree with
the figure?]].
For those of you who
did not submit the quiznot, you can try problems from older test 2 exams,
which all have answer keys. But you have to do it yourself first, then look
at the answer key or it will not be of much use.

F: No Quiz. Test 2 on chapters
3,4.
read handout on complex arithmetic, exponentials
[Maple commands; the complex number i is
uppercase I in Maple];
WEEK 9[1]:

M: Lecture Notes
5.2:
The Wronskian and higher order constant coefficient 2nd order linear DEs;
online handout:
wronskian and higher order constant coefficient linear homogeneous DEs;
[Maple Wronskian]
5.2: 1, 7, [[11]],
13, 17, 23.

T:
Lecture Notes
5.3a:
higher order constant coefficient linear homogeneous DEs and complex
exponentials;
[sinusoidaldecay.mw] [Maple Wronskian
part 2]
complex root
handouts:
1) general
amplitude and phase shift of sinusoidal
functions;
2) 4 quadrant amplitudephase shift examples:
sinusoidquadrantex.mw, [[expcossin.mw]];
3)
exponentially modulated sinusoidal functions
4)
summary handout: sinusoidal decay DE example with
complex basis
5.3: 1, 5, 9, 17, 21, 23, 33;
[[23
exanple expressed in
phaseshifted cosine form,
another example:
25]];
[ignore instructions
to factor by hand or polynomial long divide: use technology for all factoring
(bad roots!)].

W:
Lecture Notes
5.3b:
higher order constant coefficient linear homogeneous DEs: repeated roots;
[Maple Wronskian part 3] [phasor?]
5.3 (higher order DEs):
13, 33(oops), 41, 48
[[optional
repeated complex
roots: 16, 20]]
[[IMPORTANT: For the
decaying sinusoidal function: y(t) = exp(2 t) (3 cos(5
t)+4 sin(5 t)),
a) evaluate numerically the frequency ω,
the period T, the characteristic time τ, the ratio 5
τ /T, the initial amplitude and the phase shift in radians, degrees and
cycles. Identify the two envelope functions. Make a plot in the natural
decay window of the function and its envelope.
b) Evaluate the initial data or state vector for this function. What
are the complex roots which give rise to this solution of a linear
homogeneous DE (r = 1/τ plus/minus i ω ) and therefore
what is the quadratic characteristic equation and the corresponding DE for
which this is a solution?]]
repeated root cases: 5.3: 11, 25 (the hw portal has very few repeated
root problems!)

F: Quiz 7 (complex root pair DE like
this quiz);
Lecture Notes
5.4: Linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillators);
handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator)
5.4: 3 [use
meter units!], 13
(maximum displacement),
[[damped oscillations: 14]] , 15,
17, 19
[[ read 23,
pdf; this is a useful application
problem]].
WEEK 10[1]:

M:
Lecture Notes
5.5: NONhomogeneous 2nd order DEQ with constant coefficients;
[mw];
handout on driven
(nonhomogeneous) constant coeff linear DEs; [final exercise on this
handout sheet: pdf
solution];
5.5: 1, 3, 4, 8,
10, 13,
17, 33, 37
(pdf) [not easy for me either!]
[not many of the book RHS driving
functions are physically interesting here;
we will not cover "variation of parameters"; the book
presentation of the method of undetermined coefficients is a recipe with
little justification (educated guess), instead the handout shows exactly how and why one gets
the particular solutions up to these coefficients].

T: Lecture Notes
5.6a: Driven damped harmonic oscillators;
handout on damped harmonic oscillator driven by sinusoidal driving function
[1 sheet printable version];
[in class example.mw pdf:
resonance
calculation side by side], maple resonance plots: general, specific];
5.6: 1,
3, 11, 13, 17 (template
for final 3 problems, HW portal wants phaseshifted cosine form for both
steady state and transient in 11, 13, phase shift between 0 and 2 Pi,
decimal solutions);
Wiki:
driven
harmonic motion,
amplitude plots:
they introduce ζ = 1/(2Q)].

W: 5.6: Lecture Notes
5.6b: Driven damped harmonic oscillators: special cases;
handout on beating and
resonance [mw, with
optional
beating plot problem];
Online HW catchup day. Don't let these go
unfinished. Get bob's help.
optional 5.6: 1 [worksheet rewrites
this as a product of sines using the cosine difference formula:
cos(A)  cos(B) = {2 sin((AB)/2)} sin((A+B)/2), the expression in {} is the
sinusoidal amplitude envelope function [and its period is T = 2(2
π)/(AB)], look at the plot of x and +/ this amplitude function
(the envelope) together in this beating
example worksheet, then do
as explained here, repeat for the HW
problem described in this link as well as at the end of this same beating
example worksheet (be sure to plot one
full period of the envelope function to see exactly 2 beats],
We will watch the 4 minute Google linked video of resonance NOT! (but see the
engineering explanation linked PDF for the detailed explanation):
[Tacoma Narrows Bridge collapse (resonance NOT!):
Wikipedia;
Google (YouTube video)];
a real resonance bridge problem occurred more
recently: the Millenium Bridge
resonance (5 minutes).
We will watch a 9 minute YouTube video
massdampers.htm that shows a physical problem using the mathematics we
will work up to by the end
of the semester, two coupled damped oscillators,
which is a simple model for realistic damping of the swaying motion of tall
skyscrapers.
Read 5.6.23 (earthquake!):
consistent units: sec, ft, lbs; first evaluate Hookes' law constant, then
find x''+10 x = A_{0} ω^{2}
sin(ω t), convert frequency to ω
= 2 π /2.25 radians/s, set A_{0 } (=3in) = 1/4 ft. Solve. Find
amplitude of response oscillation in inches. [soln]
Ignore this: EE/Physics majors if you are interested: RLC circuit application
(floating point numbers!): RLC circuits [maple
plots]

F: Quiz 8 like 18F:Quiz 8;
Lecture Notes
6.1a: Eigenvectors and eigenvalues;
Transition back to linear algebra:
[In
class watch bob use the Maple worksheet
ExploreEigenDrag.mw to
graphically determine the eigenvectors and eigenvalues of three 2x2
matrices. Recall old handout
coupled system of DEQs and its
directionfield [motivation: direction fields for Maple help visualize eigendirections of a 2x2 matrix]
watch the
MIT Eigenvector
4 minute video [requires IE browser because of Flash; there are 6 frames which then repeat, so stop when you
see it beginning againfrom their
LinAlg course]; see the 6 possibilities of the video in this
Maple worksheet DEPlot directionfield
phaseplot template;
then play a computer game with that Maple worksheet lining up the vectors [red is x,
blue is A x,
click on matrix entries to change, click on tip of red vector and
drag around an approximate circle to see corresponding blue vector]; try
first with the default values, then try for a_{11}=0, a_{12}=1,
a_{21}=1, a_{22}=0, then try for the
matrices of 6.1: 1,2,7. See if you can guess the eigenvector directions
and the corresponding eigenvalues (all integer triangles locating the
vectors and integer eigenvalues) for
these matrices; write
down a simple representative eigenvector (with the smallest integer
components, say) and its eigenvalue that you can read off from the applet as
explained within the introductory webpage to compare with your matrix
calculations for these textbook HW problems . Then find them by the
eigenvector process.
6.1: 1, 2, 7 [1, 2, 7 mw]; 13; Ignore
polynomial division discussion, use technology for roots of polynomials!
frequency:
f or ω ?
angular frequency ω is in radians/time but
"ordinary" frequency is
in cycles/time (number of multiples of one revolution of the circle): ω = 2 π
f . Multiplying the frequency by 2 π radians converts it from a
pure number of revolutions to radians per time unit. The reciprocal of the
frequency f is just the period T = 1/f = 2 π / ω . We only
need angular frequency in our course, but when we reexpress such
frequencies in Hz (cycles/sec) we are converting to the ordinary frequency
for ease of interpretation, since radian units mean nothing to us.
WEEK 11[1]: 
M:
Lecture Notes
6.1b: Eigenvectors and eigenvalues: more (linear independence,
complex eigenvectors);
Maple
3x3 matrix eigenvector example handout [.pdf,
real3x3 .mw;
complex];
6.1: 15, 19, 23 (upper triangular so diagonal values are eigenvalues!),
(complex:) 29, 31
[do everything by hand for 2x2 matrices;
for 3x3 or higher, go thru process: use Maple determinant: AλI
= 0 (right click on matrix, Standard Operations), then solve to find characteristic equation and
its solutions, the eigenvalues,
back sub them into the matrix equations (AλI)
x = 0 and solve by rref, backsub, read off eigenvector basis of
soln space, NO POLYNOMIAL DIVISION! compare to Maple's Eigenvector result,
context menu only a].

T: 6.2:
Lecture Notes
6.2: Diagonalization;
diagonalization [now revisit
coupled system of DEQs in a handout giving a preview of what we are about to embark on:
why diagonalization?];
6.2: 3, 10, 13, 19, 25 [use Maple for det and
solve for eigenvalues, then by hand find eigenvectors, use Maple for matrix
multiplication]

W:
Lecture Notes
6.2b: Geometry of diagonalization;
on the geometry of diagonalization
and first order linear
homogeneous DE systems
Print out 3 copies of this
grid page for the homework problems;
0) [[ read
the worksheet explanation: 6.2.34
epc46634.mw]];
If you have time, find the eigenvalues/eigenvectors
by hand, if not use Maple:
1)
For the matrix of 7.2.14: A = <<32>,<34>> entered by rows,
find the eigenvalues and order them by increasing
value, then rescaling if necessary to obtain two independent integer component eigenvectors {b1, b2}, make the basis changing matrix
B = <b1b2>, use the coordinate transformations
x = B y and y = B^{1 }
x to find the new coordinates of the point
x = <x_{1}, x_{2>} =
<0,5> in the plane, then make a grid diagram with the new (labeled)
coordinate axes associated with this eigenbasis
(labeled also by the eigenvalue λ =
<value> ) together with basis vectors and
the projection parallelogram of this point; finally
evaluate the matrix product A_{B} = B^{1}A B to see that it is
diagonal and has the corresponding eigenvalues in order along the
diagonal.[multiply b1 and b2 by A to confirm the eigenvalues];
2) For the matrix of 7.4.2: A = <<54>,<45>>,
repeat with <x_{1}, x_{2>} =
<4,2>.
3)
For the matrix of 7.4.2: A = <<408>,<1260>>,
repeat with <x_{1}, x_{2>} =
<4,5>.
This is what you should have done:
diagonalizationplots.mw
No more sections of chapter 6 will be
covered.
7.1 is better done with "reduction of
order" instead of "increasing the order" to decouple
systems.

F: Quiz 9 [diagonalization of 3x3 matrices,
one real, one complex;
just the
linalg part of 18F/16F/15S: quiz 9 for the real
case; this
example for the complex case]
see:
transition from a complex to a real basis of a
solution space [complexplanecoords.pdf,
complexplanecoords.mw]
Lecture Notes
7.3a: 1st order linear homogeneous DE systems: real
eigenvalues:
the
geometry of diagonalization and first order linear
homogeneous DE systems;
(2d example: real eigenvalues [phaseplot]),
7.3: 3, 5, 7,17, 20, 23.
WEEK 12 [1]:
in progress 
M:
Lecture Notes
7.3b: 1st order linear homogeneous DE systems:
complex
eigenvalues:
[example 2]
7.3:
15, 25; [[26]].
handout
on 1st order linear
homogeneous DE systems (complex eigenvalues) (2d example: [phaseplot]);
[[optional reading: Find the general solution for the DE system x ' =
A x
for the matrix A = <<04>,<40>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,0> (solution
online:
.pdf);
Repeat the process for problem 7.3.11: x ' =
A x
for the matrix A = <<12>,<21>>,
initial conditions x(0) = <0,4> (solution
online: .mw,
.pdf)]];

T:
Lecture Notes
7.3c: nonhomogeneous 1st order linear DE systems, etc;
[epc483ex2.mw,
epc483ex2secondorder.mw]
2x2
examples handout [.mw];
handout on extending eigenvalue
decoupling
to nonhomogeneous and special second order constant
coefficient DE systems
8.2:
1,
11 (this is the 0 eigenvalue case,
so a constant driving function requires a constant times t for the trial
function, OR just integrate the decoupled first order DE using the
standard first order algorithm, see page 2 or today's lecture).

W:
Lecture Notes
7.3d: mixing tanks, etc; [example
4 closed system, complex eigenvalues; open
system, real eigenvalues;
driven case]
(compartmental analysis: Pharma,
environmental and epidemiology modeling
applications);
7.3:
33 (open, real),
37 (closed,
complex); 8.2:
15 (driven open 2 tank, like lecture example
yesterday).

F: Begin in class Test 3: 1) mass spring
system: IVP plus explore resonance; 2) complex
eigenvector DE system, 3) real eigenvector DE
system; due Tuesday midnight;
please
read the short instructions on the test and the long
instructions online; this is not a collaborative
effort; start in class;
here is a stripped
down example [read it carefully]
for the direction field and solution plot versus t,
all you need for problem 3.
Remember
complexscalarmultiplication.pdf
WEEK 13[1]:

M: use class time to keep working on Test 3.
After roll call
(because he likes hearing your voices) bob
will go to the Office Hour Anytime Zoom,
using breakout rooms for confidential
questioning.

T:
Test 3 due midnight
tonight;
Lecture Notes
7.5a: mass spring systems; [figure8curve.mw];
handout on 2 mass
spring systems: theory plus
worked examples;
7.5: 3, 5, 9 [Maple quick check template:
epc475_17.mw,
hand step details:
epc4743_9brief.mw]

W:
Lecture Notes
7.5b: mass spring systems;
7.5:
11, 13 [Maple
quick check template:
epc475_17.mw]
[commonperiod.mw]
[n equal mass, equal spring constant
systems are special:
triadiagonal matrix,
Toeplitz matrix]

F:
Lecture Notes
7.5c: mass spring system: 2axle car model:
7.5:
25, 27 [Maple
template to evaluate the DE system].
WEEK 14[1]: 
M:
Lecture Notes
7.5d: mass spring system: reduction of order and damping:
no common period example filling a parallelogram
built from the eigenvectors and natural mode
amplitudes:
Maple Exploration exercise;
[solution];
<<< exercise redone with the new matrix and
adjusted for the new parameters [plot]
homework:
by
hand calculate undriven damped initial value problem
solution and the undamped rectangle containing the
undamped solution curve in the phase plane, as
explained in the Maple worksheet
[handout on reduction
of order with exercise].

T:
Lecture Notes
7.5e: mass spring system: resonance:
resonance in multidimensional systems
[resonance addition to yesterday's
exercise; figure8system play];
catch up on your online homework;
check out past
final exams.

W: bob takes a breather; forgot to press record for
section 005 but just introduced this problem:
breakout room
problem
7.5.14 (scroll to the bottom of the page);
finish for homework and go
over
18F final exam.

F:
Test 3 answer key is online, I hope to grade it
this weekend;
[dynamic damping soln:
pdf, mw];
earthquake response analysis
(toy model) for fun
[n > 2 means large n too
where
technology can implement our methods used in HW for n
= 2 [not on the final exam, :) ].
WEEK 14: 
@
M: final day;
what comes next?;
final exam discussion (one problem, concrete
numbers, no Maple plots);
CATS time for those who
have not yet done them for this class.
I am
working on Test 3!
 scroll up for current day
Weeks 2 and 3 thru 4: come by and find me in my Zoom office hour, tell me how things are
going.
This is a required visit. Only takes 5 minutes or less.
If you are at all confused, try to do this way before Test 1 in week 4.
*MAPLE homework log and instructions [asterisk
"*" marked homework problems]
Test 1: week 5
Test 2: week 9
Test 3: Take home outin week 12
Final Exam: take home released monday Nov 30, due by Friday, Dec 4.
FINAL EXAM:
pandemic conditions make this a "take home" format like the tests
so these time
slots are irrelevant:
270504 (10:30class): MWF 10:20 AM Mon, Nov 30 10:45  1:15
270505 (11:30class): MWF 11:30 AM Fri, Dec 4 2:30  5:00
instead the exam will be released Monday November 30
and due any time during
the week through Friday Dec 4, but sooner is better for grading!
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
4dec2020 [course
homepage]
[log from last time taught]
after you solve this
problem yourself, consult both [pdf,
.mw];
please read Maple worksheet to see
the power of mathematics to solve all such problems at once;
please read the PDF to see how to convert the word problem line by
line to a mathematical problem
[If you are at all confused, here is a
completed worksheet for the in
class practice example above.]
[If you are feeling ambitious, you could also edit the Maple version
of the example to see that your hand work is correct.]
soln: pdf,
mw].