MAT2705 09F homework and daily class
log
Your homework will appear here each day as it is assigned, 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, numbered consecutively below and labeled
by the (first initial of the) day of the week.] Usually it will be summarized on the white board in class, but if
not, it is your responsibility to check it here. 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).
- M: GETTING STARTED STUFF. By Wednesday, August
26, e-mail me [robert.jantzen@villanova.edu]
from your OFFICIAL Villanova e-mail account (which identifies you with your
full name) with the subject heading "[MAT2705-XX]", where XX is your
section number 02 or 06, telling about your last math courses, your comfort level with graphing
calculators (can you do symbolic derivatives and integrals on your graphing
calculator if not in Maple?) and computers and math itself [sophomores only: how much experience you have with MAPLE (and Mathcad
if appropriate) so far], why you chose your major, etc.
Tell me what your previous math course was named (for sophomores: Mat2500 = Calc 3 or
Mat1505 = Calc 2).
[In ALL email to me, include the string "mat2705"
(no spaces) somewhere in the
subject heading if you want me to read it. I filter my email.]
In class if you brought your laptop (otherwise log in to the computer at
your desk):
1) log on to your computer and open
Internet Explorer.
(IE allows you to open MAPLE
files linked to web pages automatically if MAPLE is a local application,
while in citrix Maple, you must save the file locally and then open in it MAPLE using the
File Open task or using the Open URL selection.)
2)
log on to the Villanova home page
MyNova
in Internet Explorer (click
on the upper right "login" icon and use your standard VU email username and
password) and check out our classroom site, and visit the link to
my course homepage from it
[
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
3)
Open Maple 13 Standard (red not yellow icon) from the Start Menu Program
listing under Math Applications
[or click on this maple file link:
Maple example file]
[Using the File OPEN URL command in Citrix Maple, with this URL (right
click, copy shortcut) will show
you a Maple file of how to pick the viewing window for realistic exponential and sinusoidal example functions.
Clicking on the !!! icon on the tool bar with execute this worksheet
and restore the output which has been removed to make the file smaller. It
will also automatically open the two subsections, which can be closed by
clicking on the triangle at the section title. Try this at home to make sure
you can access citrix Maple.]
4) bob will quickly show you the computer environment supporting
our class. And chat up a bit the course. [cell phones on vibrate or off: bob
will give you his cell phone number in class, not on the web!] [maybe he
will try to impress you with this gee whizz!
Maple demo]
Afterclass:
5)
log on to My Nova and go to WebCT and look at the 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.
[This is the only part of WebCT we will use this semester.]
In the Student Tab, click on the double person icon to the right of our
class line to get to our photo class roster and click on my home page URL
under my photo. Click on our class URL there.
Check out the on-line links describing aspects of the course (no need yet to look at the
MAPLE stuff). Fill out your
paper
schedule form (get a copy in class to fill out or print it out
back-to-back earlier
to fill out in advance and bring to your first class already done; consult the
dorm abbreviation
list)
to return in class Tuesday.
[You can
drop by my office St Aug 370 (third floor, Mendel side, by
side stairwell) to talk with me about the course if you
wish and to see where you can find me in the future when you need to.]
6)
Download Maple 13 and install it on your laptop when you get a chance
(it takes about 15 minutes, I will help you in my office if you wish). If
you have any trouble, email me with an explanation of the errors.
[You can always access and open Maple 13 Standard in citrixweb
on any Windows computer anywhere through Internet Explorer but the
first time you use citrix on a computer, you must download and install the
client from the link on that web page, but the local copy of Maple is
preferable.]
You are expected to be able to use Maple on your laptop or Windows
Maple on class computers when
needed. We will develop the experience as we go.
7) Read computer classroom
/laptop etiquette.
Then read the first paper
handout: algebra/calc background
sheet.
8)
Homework Problems:
1.1: 3, 5,
13, 33 (this links to a PDF scan of the HW problems from the book if you have
not yet purchased it, edition 2, maybe the problems have not changed);
[ "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.
Get acquainted with Maple
Standard DE entry and "odetest" for problems 3,5,13 [even
33!]
9) Fill out the paper schedule form bob handed out in class. [see
handouts]; use the 3 letter
dorm abbreviations.
- W: return your schedule forms at the beginning of
class;
[feed the URL of this maple file link:
Maple example file into Maple in class following
bob's lead];
1:1: 7a [i.e., only check y1(x)], 23;
formulating DEs: 27, 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],
35, 36.
- Th: Quiz 1;
1.2: 1, 5, 15,
19 [Hint: write piecewise linear function from graph: v1(t) for first expression,
v2(t) for second expression, solve first IVP
for x1(0) = 0, second IVP for x2(5) = x1(5)],
25 (like lunar landing problem,
see example), 39, 40 (ans: 2.4mi),
43 (pdf)
[convert to appropriate units!].
- F: course info handout;
quiz answer key online;
on-line handout:
initial data: what's the deal?;
1.3: 3:
[hand draw in all the curves on the printout supplied by bob (rotated and
scaled to fit option on Print), maple
version],
8* : our first Maple HW problem: read the
Maple HW instruction page, then try the Maple direction field
command in this linked template to reproduce the
two given curves by using
two appropriate initial data points [Hint: notice that the y-axis
crossing points are integers!] Then include all the initial data points of
the red dots (some half-integer values); this is also described on p.29
for both Maple and Matlab, but the above template already has the
appropriate section copied into it for you; if you can find a partner to
work with in time fine, otherwise try this yourself and find a partner for
future problems, then merge your work [will not be due until we
finish chapter 1];
being careful:
11, 15 [note Dyf is just df/dy above Eq (9) in the
text]; 27
[don't worry about making a diagram];
a freebie: numerical problems caused by
vertical tangent line (figure 1.4.2).
WEEK 2:
- M:
1.4: 1, 5; 21, 25, 27, 29.
- W: handout on
exponential behavior/ characteristic time [read
this worksheet explicit
plot example];
1.4: 45, 47 [recall answer to 1.1.35, use units of
thousands of people, assume no one has heard the rumor at t = 0: N(0) = 0 (an
approximation in units of thousands; really N(0) = .001 would be one person
starting the rumor, or should it be two?)],
65 [pdf,
.mw];
Optional challenge problems:
68 (PHYS: parametric cycloid solution, obvious typo in text:
θ = 2 a should instead be
θ = 2 t)
or 69 (ME: suspension cable catenary: hyperbolic cosine)
or Torricelli's law section and 64 (ChmE: ans r = 1/35 in).
- Th: class list contact data sheet handout (not web
published) to help form partnerships;
check quiz 1 grades in BlackBoard gradebook accessible through
our MyNova ;
handout:
recipe for first order linear DE;
1.5: 3, 7, 11, 21; 27;
21*[check 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]
> y ' = x y , y(0)=1
here is a cute solution test command:
> odetest(sol,deq) [this returns 0 if LHS - RHS of DE is 0]
- F: Quiz 2;
how to
understand the DE machine;
1.5: 17, 26 [ans: x = ½
y -2 +Cy -4],
29 [use the fact that the derivative of
the function defined by the integral is the integrand including all
multiplicative factors],
31
[show by differentiation and backsubstitution into the DE, not by rederiving it],
32 [find the constants by substituting the trial solution into the DE and
choosing the values of A and B to make the DE satisfied: put all terms on LHS
and set coefficients of the sine and cosine separately to zero],
41 [Hint:
Δt in years is the fraction of the year, so S(t)
Δt is the approximate income during this time interval, 12 percent of which
contributes to ΔA, while the interest contribution is the annual interest rate
6 percent prorated by multiplying by the fraction Δt for the interest
contribution over this shorter time period, applied to the current value of
the retirement amount A(t): 6 percent of A(t) Δt; adding these and dividing by
Δt, taking limit to get dA/dt gives the linear DE [click
here only if you have tried to get the DE and could not or its solution did not give
the correct number].
41* [use the dsolve template to solve this DE with its
initial condition; evaluate answer to question of word problem to a floating
point number and give the result in dollars and cents in a comment, then think
about significant figures and give a response with fewer significant digits].
When you are satisfied, compare with
retirement.pdf.
WEEK 3[-1]:
Weeks 3 thru 4: come by and find me in my office, tell me how things are
going.
This is a required visit. Only takes 5 minutes or less. Come say hi!
Labor Day Weekend!
REMIND BOB to match up people who need Maple partners.
- W: 1.5: 37
(Use
Eq. 18 in the book; what is the final concentration of salt?);
[this mixing problem (online
handout) is an
example of developing and solving a differential equation that models a
physical situation; for another word problem example,
see this]
1.R(review): classify the odd problems 1-35 as:
separable, linear in y
(as unknown),
linear in x (as unknown), some combination of these three, or NOTA (none of the above), for
example dy/dx = y/x is all three and can be solved in three different ways;
don't solve them but:
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 u-substitution
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 results.
- Th: Quiz 3 linear 1st order ODEs;
handout on solution of logistic DEQ
[directionfield,
integral formula];
2.1: 5 [go thru the solution steps
just this once with concrete numbers, remember the
integral formula, follow the steps of the book or handout derivation of the solution,
compare with the final logistic formula; use the formula for the remaining HW
problems],
9, 15 [note that by definition Bo = kMPo and
Do =
-kPo2 , and these are easily solved for M and
kM in
terms of Bo, Do, and Po to get the other two
logistic curve parameters needed to solve problem 16],
16 [ans: P = .95 M
after 27.69 months].
Optional Light Reading 1.5: Application: see how a slight generalization of
the directionfield example 1.3.3 to
include an initial time parameter and time and temperature scale parameters
has a useful physical application on pp.56-58. No need to study it, but
reading through it quickly gives an idea of how one can use a simple linear DE to
model an interesting problem.
- F:
handout on DE's that don't involve the ind
var explicitly;
2.1: 6
[use handout integration formula with sign reversed, or use technology
for integral and combine log terms;
answer: > dsolve({x'(t) =3 x(t) (x(t)-5), x(0)=2},x(t))
(copy and paste into the input line);
plot this
solution with technology and choose a horizontal window (negative and positive t
values) in which you see the reversed S-curve nicely, be ready to give
the approximate time interval for an appropriate viewing window
in class],
11 ["inversely propto sqrt": use β = k1/P^(1/2), δ =
k2/P^(1/2) in eq.(1)],
23 [Note dx/dt = 0.004 x (200-x) so k = 0.004,
M=200 for logistic curve
solution formula],
30, 31 [ans: find βo = 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)].
Did you stop by bob's office yet? TEST 1 next Thursday.
5:30pm MLRC voluntary Test 1 problem
session Tuesday.
WEEK 4[-1]: maple1.mw due this week following instructions on maple hw
page
- M:
air resistance handout [comparison
of linear, quadratic cases; numerical solution];
2.3: 1, 2, 3, 9 [remember weight is mg, so mg = 32000 lb determines
m = 1000 in USA units],
22 [this is the tanh case: vterminal = 20.7 ft/s, t = 8 min 5 s][solution,
.mw];
Optional 23: comparison of two cases like 22.
Tuesday 5:30 MLRC voluntary problem session.
- W: 2.4: 1 (do by hand and calculator, can check with euler.mw if you wish;
optional read about improved Euler): due Friday after test;
maple2.mw:
instructions for this maple assignment.
[due this coming week Friday to Friday; make sure you have at least one
partner and follow the submission instructions.]
- Th: Test 1.
- F: Test 1 answer key online;
handout: 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: 1, 5, 7; 9, 15, 31, 33, 34.
WEEK 5[-1]:
- M:
handout on RREF
(Reduced Row Echelon Form, section 3.3);
3.2:
1, 3,
7; 11, 13, 15 (do a few by hand, then you may use step-by-step
row ops with MAPLE or
a calculator for the rest;
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: maple1 is past due; maple2
is due by this weekend;
handout on
solving linear systems example;
we always want to do a full reduction, not the partial Gauss
elimination reduction also described by the textbook;
in class open this Maple file with a partner
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, 7, 11, 17, 23 = 3.2.13;
19* [use the tutor and record your step by step reduction, annotating each
step as in the
row ops with MAPLE worksheet (here are
some more examples rowredex0.mw); then check with ReducedRowEchelonForm
or by right-clicking and choosing from the menu (solvers and forms)].
To solve a linear system directly in Maple, input augmented matrix with the
palette or Matrix command and these 4 lines will do it:
> with(Student[LinearAlgebra]):
[you can load this from Tools, Load Package,...]
A:=Matrix([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);
[you enter matrix using palette instead]
ReducedRowEchelonForm(A);
[you can right click reduced this: solvers and forms]
BackwardSubstitute(%);
To step by step reduce a matrix (max 5x6) and solve the system do (pick
Gauss-Jordan):
> LinearSolveTutor(A)
Matrices for doing homework with Maple
- Th:
3.2: 22
[solve using row
ops on augmented matrix to rref form, then by hand backsubstitute, check
answer with Maple];
3.3:
29 = 3.2.19 [solve
using row ops on augmented matrix to rref form, then by hand backsubstitute];
29* [now check this same problem solution by solving with
ReducedRowEchelonForm, BackwardSubstitute
of augmented matrix after inputting the Student Linear Algebra
package from the Tools menu];
MAPLE
chemical reaction problem*;
word of the day: can you say "homogeneous"?
- F: Quiz 4 on using row reduction to solve
linear systems;
3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
Matrix algebra is easy in Maple.
WEEK 6[-1]:
- M:
3.5: 1; 9, 11; 23 [this is a way of solving 3
linear systems with same coefficient matrix simultaneously];
30 [just substitute for the inverse and then drop parentheses and multiply
out: (ABC)-1 (ABC) = ...],
32 [multiply on the left by A-1].
Matrix multiplication and matrix inverse, determinant, transpose [or
right-click and use menu]
> A B [note space between
symbols to imply multiplication]
> A-1
> |A| [absolute value sign
gives determinant]
> A%T
- W: read on-line handout:
determinants and area etc;
3.6: (determinants abbreviated: forget
about minors, cofactors, only need row reduction evaluation to understand)
13, 17, 21
(plug in just so you've done it once);
17* [record your Gaussian elimination
steps in reducing this to triangular form and check det value against your
result];
3.5. Application:
7* from textbook on p.201,
using the inverse matrix as described on p.200,
be sure to answer the word problem question with a text comment;
And also the question: are these sandwiches cheap or not?
- Quiz 5 on vector form of solution of
linear homogeneous system, using inverse matrix to solve a square linear
system;
Th: 4.1: 1; 5, 7; 9; 15, 17; 19, 23; 25 (use technology in
1) evaluating
the determinant or 2)
rightclick menu to the ReducedRowEchelonForm
needed
for these problems);
[HW row reductions];
hand out on the interpretation of solving linear
homogeneous systems of equations: A x = 0 [optional
read:
visualize
the vectors]
maple3.mw assignment
is complete; due next week or at least during the week after break if you have too
much midterm stuff going on; get it over with early, catch
up with others if not done yet
[blowing off maple means subtracting 10 points from your cum for midterm
grades: maple1 and maple2 only for midterm grades].
- F:
handout on nonstandard coordinates on R2 and
R3 [goal: understand the jargon and how
things work];
1) Using the completed first page coordinate handout
made for the new basis {[2,1],[1,3]} of the plane: graphically find the new
coordinates of the point [4,7], then confirm using the matrix multiplication
by the inverse matrix of the coordinate transformation given there. Then
find the old coordinates for the point whose new coordinates are [2,2].
2) Using the previous handout as a guide, first fill in the blanks on the
sheet handed out in class for the
basis transformation matrix and its inverse, and then for the change of
coordinates, and then use either the matrix or scalar equations of the
change of variables to calculate the other coordinates for the two given
points.
Then following the example, using a ruler and sharp
pencil, make a new -2..2,-2..2 coordinate grid associated with the new basis
vectors {[1,1],[-2,1]} and graphically represent the point [x1,x2]
= [-5,1] and find
its new coordinates [y1,y2] using the grid and confirm that this agrees with your
previous calculation. Similarly read
off the old coordinates [x1,x2] of the point whose new coordinates are
[y1,y2] = [2,-1]. Confirm that these
agree with your calculations. Put your name on it and bring it to class.
[You could also edit the Maple version
of the example to see that your hand work is correct.]
WEEK 7[-1]: Maple 3 due this week, but make sure Maple1,2 upgraded to 2
for midterm grades.
- M:
study the handout on solving linear systems revisited;
[note only a set of vectors resulting from 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, 11; 15, 19;
Look at the solution to the system 3.4.21 on p.186 (back of book p.724): what does it tell us about
the 5 vectors (namely the columns of the coefficient matrix of the linear
system) in R2? How many
independent relationships are there among the 5 vectors? How many vectors are
independent? Is this obvious if you look at the explicit components of these
vectors? [no! but if you realize that no 2 vectors are proportional, and no
more than 2 vectors can be independent in the plane, of course the answer is
obvious, this is the power of reasoning].
- W: handout on
linear combinations, forwards and
backwards [maple to visualize];
what does the solution of a
nonhomogeneous system of equations say about the column vectors in the
augmented matrix?
4.3: 1, 3, 5, 7; 9, 13; 17, 21; 23.
- Th: Quiz 6 (see 08f Q5);
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 (2x2 dets are easy by hand!)].
- F: handout summarizing
linear vocabulary;
[maple
illustration of how to describe a plane in space using a subspace basis].
[optional on-line only extra handout:
linear system vocabulary]
Maple3 due by this weekend or after break
but make sure Maple1,2 upgraded to 2 for midterm grades by Monday after
break.
Fall Break. :-) Enjoy. Be safe in your travels.
.
Week 8[-1]: test 2 on chapters 3, 4 in next week 9: Th
- M: catch up on 4.4 hw,
any questions?;
Part 2 of course begins: transition back to DEs:
read the 4.7 subsection (p.279)
on function spaces and then examples 3, 6, 8, 9; read the worksheet on the
vector space of quadratic functions [quadratics.mw,
pdf];
4.7: 15 [Hint: solve c1(1 + x) + c2(1 - x) + c3(1 - x2) = 0, for unknowns c1,c2,c3;
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],
16, 17 [same approach].
- W: Memorize: y ' = k y < -- > y = C e k x
y '' + ω2 y = 0 < -- > y =
C1 cos(ω x) + C2 sin(ω x)
5.1[up thru page 295, plus Example 7]: 1, 3, 5, 9; 13,
17;
begin maple5.mw:
9* [use the 2nd order dsolve template to
solve the 2nd order IVP as a check].
> deq:= y'' + 3 y' + 2 y = 6 ex
> inits:= y(0) = 4, y'(0) = 5
> solgen:=dsolve(deq,y(x))
> solivp:=dsolve({deq,inits},y(x))
or right click on output of (use solve DE interactive, Solve
Symbolically, Solve, Quit):
>
y'' + 3 y' + 2 y = 6 ex
, y(0) = 4, y'(0) = 5
- Th: handout on sinusoidal example;
5.1[after page 295]:
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]
5.2: 1,
11;
5.1: 49*[check your IVP solution for the highest curve in Figure 5.1.6
(highest for x > 0) using
the 2nd order dsolve template and plot it together
with the horizontal line y = 16/7 claimed by the back of book answer as the highest value of
that function; recall how to plot multiple functions in the same plot:
> plot([<expression_for_y(x)>,16/7], x=a..b, y=c..d, color=[red,blue]);
the color option is useful in distinguishing two functions when it is not
already clear which is which; is 16/7 the peak value of your
solution?].
- F: Quiz 7 (like quiz 6 before);
read handout on complex arithmetic, exponentials
[maple commands; the complex number i is
uppercase I in Maple];
5.2:
13, 17, 21, 26 [this problem
is for fun now, but later
we learn a method to attack it];
read the optional on-line handout on
visualizing the initial value problem (IVP)
if you wish to see the interpretation of solving the IVP.
WEEK 9[-1]:
- M:
5.3: complex roots:
8 [ans: y = exp(3x)(c1 cos(2x) + c2 sin(2x))] , 9; 17;
22,
23 (express in phase-shifted cosine form,
see handout on the
amplitude and phase shift of sinusoidal
functions);
use technology to solve for roots of polynomial equations (or
right-click on eqn output, Solve, Solve) :
> solve(r2+6 r+13 = 0) (in
general, finds all roots exactly up to fourth degree and sometimes higher
degree if lucky)
[Note: > fsolve(...) returns all numerical roots of a polynomial].
TUESDAY: Test 2 MLRC 5:30 voluntary problem session?
TUESDAY since my wife's birthday is Wednesday.
- W: 5.3 higher
order DEs [ignore instructions to factor by hand or
polynomial long divide: use technology for all factoring]: 11, 13;
25, 33; 39;
49 ;
49* [check your solution with the higher
order dsolve template, and also check the IC linear system solution as in
the template, edit away all stuff not directly relevant to your new problem].
handouts on exponentially modulated sinusoidal functions [maple
videos showing different phase shifts and frequencies];
do the short problems at the end of each sinusoidal handout after studying
them (to get the phase angle right). [soln]
- Th: Test 2 on chapters 3,4.
- F:
5.4:
1, 3, 13,
14 [b) the solution once put in the form
x = A exp(-Kt) cos(ω
t
- δ) has the "envelope curves"
x = ± A exp(-Kt) ] ,
17;
23 [pdf].
handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator)
[on-line only: examples in
nature].
WEEK 10[-1]:
- M:
5.4: 21 (solve by hand),
21* [solve the two DEs with the same initial
conditions using Maple and plot the two solution curves together, but also plot the corresponding third solution with pure damping but no
oscillator as in the template];
handout on driven
(nonhomogeneous) constant coeff linear DEs
[complete final exercise on handout sheet, Maple: *check this with dsolve;
pdf solution];
5.5: 1, 3, 9
[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].
- W: T: handout on damped harmonic oscillator driven by sinusoidal driving function
[class example, Maple specific example]
[maple resonance plots: general ,
specific (frequency much higher than
resonance)];
5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85
- (6 cos(3x)+7
sin(3x))/85]; [not many of the book driving functions are physically
interesting here]
5.6: 8. Ignore the book instructions and solve this problem by hand for the
initial conditions x(0) = 0 = x'(0). Read the
worksheet about the interpretation of
the solution.
- W: [ >
x, -x, x sin(x) Enter and right click on output
(choose Plot Builder, 2D Plot)
to plot together to see oscillation and its envelope. or use bob's templates
in each problem.]
handout on beating and
resonance
[Tacoma Narrows Bridge collapse:
engineering
explanation;
Wikipedia;
Google (choose a
You-Tube video)];
maple5.mw due next week (5 problems now complete);
5.6: 1 [rewrite as a product of sines using the cosine difference formula:
cos(A) - cos(B) = {-2 sin((A-B)/2)} sin((A+B)/2), the expression in {} is the
sinusoidal amplitude, look at the plot of x and +/- this amplitude function
together, then repeat for the HW
problem * at the end of the beating example worksheet and include only that HW problem
section in your maple5 worksheet],
11 [convert steady state solution to phase shifted cosine at
end of
calculation], 17 [pdf],
23.
Test 2 back. Make sure you study the answer key.
- F: Quiz 8 [similar to
07F Quiz 8 but not just changed numbers];
study the pdf solution of 5.6.17:
17 [pdf];
Transition back to linear algebra:
Fun.
Watch the
MIT Eigenvector
3 minute video [there are 5 frames which then repeat, so stop when you
see it beginning again---from their
LinAlg course]; see the 5 possibilities of the video in this
Maple DEPlot
phaseplot template;
then play a computer game with the
Duke U applet 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 a11=0, a12=1,
a21=1, a22=0, then try for the
matrices of 6.1: 1,2,3,9. See if you can guess the eigenvector directions
and the corresponding eigenvalues (all integer triangles locating the
vectors and integer eigenvalues); 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.
WEEK 11[-1] maple5.mw is due any time next week or earlier
- M:
turn in weekend take home quiz 8;
recall
coupled system of DEQs [now revisit
it in a handout giving a preview of what we are about to embark on];
6.1: 1, 9 , 13, 19; 27, 25;
[do everything by hand for a 2x2 matrix; go thru process: use determinant: |A-λI|, solve to find characteristic equation and eigenvalues
of a 3x3 or higher dim matrix, NO POLYNOMIAL DIVISION!].
- W: For the matrix A = <<1|4>,<2|3>> entered by rows,
from the two independent integer component eigenvectors {b1 = <1,1>,b2 =
<-2,1>}, make the basis changing matrix
B = <b1|b2> whose columns are these vectors, use the two coordinate transformations
X = B Y and Y = B-1X
to find
the new coordinates of the point [x1, x2] = [-2,4] in the plane and to find the point
whose new coordinates are [y1, y2] = [2,-1]; finally evaluate the matrix
product by hand AB = B-1AB to see that it is
diagonal and has the corresponding eigenvalues in order along the
diagonal.[multiply b1 and b2 by A to see what multiple
you get in each case---the eigenvalue];
[solution: remember this? (now the geometry of diagonalization)];
6.2: 1, 9 ("defective", see
direction field plot); 13 (upper
triangular!, det easy), 21 ("defective"); [use Maple for det and
solve for eigenvalues, then by hand find eigenvectors]
34 [just write out the quadratic characteristic eqn and solve with the quadratic formula, think about
real distinct roots; Delta is the discriminant; what if b = c
when the matrix is "symmetric"?].
- Th: handout on the geometry of diagonalization
and first order linear
homogeneous DE systems (2-d example: real eigenvalues [phaseplot]);
1)
Repeat this handout exercise for the matrix of 6.2.1 (namely
A = <<5|-4>,<2|-1>>
input by rows) and the initial condition x(0) = <0,1>;
2)
solve the 3x3 system x' = A x for the matrix of
problem 6.1.19, with initial condition x(0) = <2,1,2>;
3)
find the eigenvalues and matrix B of eigenvectors
for the matrix A = <<0|4>,<-4|0>>
(input by rows) and verify that AD = B-1A B is diagonal.
- F: Quiz 9 on deriving eigenvalues/eigenvectors;
handout on 1st order linear
homogeneous DE systems (2-d example: complex eigenvalues [phaseplot]);
handout on transition
from a complex to a real basis of a linear DE solution space;
Find the general solution for the DE system x ' =
A x
for the matrix A = <<0|4>,<-4|0>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,0>; (solution on-line:
.pdf)
Repeat the process for problem 7.3.11: x ' =
A x
for the matrix A = <<1|-2>,<2|1>>,
initial conditions x(0) = <0,4>
[if you like you can quickly edit the DEPlot
phaseplot template to reproduce the back of the book diagram].
WEEK 12[-1]:
- M: handout on
1st order linear
homogeneous DE systems (2d example: purely imaginary eigenvalues);
using the
eigenvector technique, find the general solution for the DE system x ' = A x
for the matrix A = <<1|-5>,<1|-1>> (input by rows) and then the solution
satisfying the initial conditions x(0) = <1,2>; note the solutions are
easily obtained with dsolve (*check
your hand solution this way!);
Read 7.3 [especially compartmental analysis example 4, which has complex
eigenvectors (more examples)];
do 7.3: 37 [closed 3 tank system with
oscillations (explanation,
text example), plug into Eq(22), solve by
eigenvector method: pdf,
.mw].
- W: Read 7.1 excluding examples 5-7 (these are the opposite of reduction of order, not
needed);
read
handout on reduction of order with exercise
and do exercise begun in class; [pdf
solution]
(optional
on-line handout on phase spaces if you
want to understand how this is connected to velocity and momentum spaces);
the rest of the homework consists of very short problems, no solving required,
just rewrites, to make sure you understand matrix notation:
7.1: 1, 8 [first let = [x1,x2,x3,x4] = [x,y,x',y'], then re-express the 2
DEs
replacing x1" by x3' and x2" by x4', adding the definitions
x1' = x3, x2' =
x4, then write the 4 DEs in matrix form x ' = A
x + F],
Note: substituting x = B y , leads to B y
' = A B y +F hence y ' = B-1A B y + B-1F
so we can also solve nonhomogeneous linear DE systems in the same way,
adding the new components B-1F of the driving
vector function to the new decoupled DEs;
7.2: 5, 9. This will be
on the Friday quiz.
- Th: summary handout on
eigenvalue decoupling so far;
handout on extending eigenvalue
decoupling (to nonhomogeneous case, and second order) [example];
solve x''(t) = A x(t) + F(t),
where x(t) = <x1,x2>, F(t) =
<0,cos(t)>, A = <<0,4>,<4,0>>, x(0)=<1,0>, x'(0)=<0,0>
[solution: .mw, eigenvector
details].
- F: Quiz 10 (last one!);
read 7.2. do 7.2.23 (solving initial conditions);
make sure Maple5.mw is submitted soon as well as any back upgrades you want.
WEEK 13: Take home Test 3 out sometime this week
after T-Day for one week:
- M: MLRC Test 3 problem session 5:30pm
Hw solution: [pdf,
mw]
handout on 2 spring 2
mass system; read handout so you understand, and then solve the stated
initial conditions for the system initially at rest at equilibrium to obtain
the result on the handout (lower left corner).
T-Day!
Be safe in your travels.
Please read the Test instructions
first.
Here is the test 3 ../testquiz/09f/27509f31.pdf
[corrected]
Remember, no collaboration.
[You will need the Maple DEPlot
phaseplot template, or the DEplot example from the last test 3. If you have any Maple problems while doing this
test, email bob immediately with your worksheet attached and a clear
explanation of the problem, or if you just need to ask how to do something,
ask. Here is the stripped down example
for the direction field and solution plot versus t.]
- M: Wow, how did the semester go by so fast?!
I will provide a printed copy of the test. You may use paper of your
choice to work it.
First Test correction: 1f) Does... A(6). Delete everything from
"Does" to "A(6)" since it is not relevant to this test, result is now: "Does
A(6) agree with A from part d)?".
Note: part 3d) does not say you have to find the solution by hand in
order to plot it. You may use the Maple solution.
- W: Bring your work on Test 3 to class today too.
- Th: Quiz 10 answer key will be on-line this afternoon.
- F: Take home Test 3 return? (I don't want you to be overloaded by work
due this week in all your courses. If you need an extension over the weekend
or even longer because you were
in this situation, ask by email on Thursday-Friday.)
Reduction of order revisited: two steps (introduce extra equations defining new
variables for derivatives up to one less than order of DEs and rewrite
original DEs in terms of their derivatives to get first order eqs; then
rewrite in matrix form for old plus new unknowns stacked together). [maple
example]
Back to coupled damped harmonic oscillator systems: bring paper
handout on 2 spring 2
mass system; read handout so you understand, and then solve the stated
initial conditions for the system initially at rest at equilibrium (starting
from the given general solution) to obtain
the result on the handout (lower left corner).
If necessary we can do this in class together on Monday. [.mw]
WEEK 14:
- M: Test 3 returned? If a further extension is needed, just ask;
we complete the homework problem in class and start together a group
exercise with 2 spring, 2 mass system and gravity, easy
numbers; complete for homework. [.pdf,
.mw] Read 7.4.
- T (F): Section 7.4 discusses resonance for
coupled spring systems;
repeat the previous 2 mass 2 spring system but with gravity replaced by a
sinusoidal pull on the second mass:
solve x''(t) = A x(t) + F(t),
where x(t) = <x1, x2>, F(t) =
<0, cos(2t)>, A = <<-5,2>,<2,-2>>, x(0) = <0,1>, x'(0)
= <0,0>;
then find the response (particular) solution for the previous system for a
general frequency ω in place of the value 2 in the previous problem;
rewrite the system in first order form as a first order DE for the 4
component state vector = initial data vector = < x(t), x'(t)>
(necessary to make the system realistic with a damping term);
[solutions: (pdf,pdf2,
.mw)].
- W: 7.4: (3 spring 2 mass system) 3, 9 (undriven/driven 3
spring system)
[7.4.9 continues 7.4.3; can use direct substitution of y1 = c5 cos(3t),
y2 = c6 cos(3t) into DEs to determine undetermined coefficients
c5,c6
for the particular ("response") solution since there is no damping term to mix cosines and sines;
also solve the IVP with inits: x(0) = 0, x'(0)
= 0: [pdf, .mw];
result is x = cos(t) X + cos(2t)
Y + cos(3t) Z, interpretation:
X = <5,5>,1:1 amp ratio tandem natural mode,
ω = 1
Y = <-8,4>, 2:1 amp ratio accordion natural mode,
ω = 2
Z = <3,-9>, 1:3 amp ratio accordion mode response mode,
ω = 3.
- Th: Teaching evaluations. Final exam remarks.[08f
final, Q10]
Final exam MLRC problem session today Thursday, 5:30pm
Final Exam Saturday (1:30-4pm) and Monday (10:45-1:15), switching
allowed just email me.
Weeks 3 thru 4: come by and find me in my office, tell me how things are
going. This is a required visit. Only takes 5 minutes or less.
*MAPLE homework log and instructions [asterisk
"*" marked homework problems]
Test 1: week 4
Test 2: week 9
Test 3: Take home out
, in week 13
Final Exam: MLRC problem session
FINAL EXAM:
[switching between these slots with permission]
2705-02: MWF/Th 11:30/1:00 class: Mon, Dec 14
10:45 - 1:15
2705-06: MWF/Th 10:30/12:00 class: Sat, Dec 12 1:30 -
4:00
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
9-dec-2009 [course
homepage]
[log from last time taught with 2nd Edition
problem numbers identical]
extras, unused stuff
.
Be safe in your travels.