MAT2705 12F homework and daily class
log
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. (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).
- M (August 27, 2011): GETTING STARTED STUFF. By Wednesday, August 29, 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]",
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. Tell me
what your previous math course was named (if at VU: Mat1500 = Calc 1, Mat1505 = Calc
2, Mat2500 = Calc3).
HINT: Just reply to the welcome
email I sent you before classes started.
[In ALL email to me, include the string "mat2705" somewhere in the
subject heading if you want me to read it. I filter my email.]
DURING CLASS (THIS IS THE FIRST DAY PLAY PART)
On your laptop:
1) Open
Internet Explorer.
(IE allows you to open Maple files linked to web pages
automatically if Maple is installed on your computer.)
2)
Log in to MyNova on the Villanova home page
in Internet Explorer (click
on the upper right "login" icon and use your standard VU email username and
password) and check out our class photo roster, and visit the link to
my course homepage from it by clicking on my home page URL under my
photo and then on our class homepage, directly:
[
http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
3)
Open Maple 16 Standard or Maple 15 if you already
have it
[or click on this maple file link:
firstdayplay.mw]
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; naahh...we'll leave this
only to the curious among us.]
AFTER CLASS (THIS IS THE HOMEWORK)
5)
log on to My Nova, choose the student tab, and go to
BlackBoard 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 BlackBoard 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 [look at the photo class roster
to identify your neighbors in class!]
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).
[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 16
if you haven't already done so 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 16 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, especially for printing purposes.]
You are expected to be able to use Maple on your laptop when
needed. We will develop the experience as we go.
7) Read computer classroom
/laptop etiquette.
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);
[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
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 to return in class the next class day.
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/x2
means y = k/x2
- W: return your schedule forms at the beginning of
class;
1:1: 7a [i.e., only check y1(x)], 23 [see
Maple plot];
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;
read the first paper
handout: algebra/calc background
sheet [online only: more rules of
algebra NOT!];
on-line handout:
initial data: what's the deal?
- Th: 1.2 (antiderivatives as DEs): 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: Quiz 1 through 1.1;
1.3:
3:
[hand draw in all the curves on the full page paper
printout supplied by bob]
11,
15-16 [note Dyf is just df/dy above Eq (9) in the
text]; 27
[don't worry about making a diagram: .mw];
Maple HW not due till the week after we finish chapter 1, there is time next
week to get a partner and ask bob questions about how to use Maple;
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].
If time:
Right click menu driven Maple intro for
DEs explained step by step in class.
Read the
course info handout. Office hours are set.
WEEK 2[-1]:
Labor Day Monday, no classes. probably no hard labor either.
- W: 1.4: 1.4 (
separable DEs ) 1, 5; 21, 25, 27, 29 [example
2: technology hint].
- Th: handout on
exponential behavior/ characteristic time [read this worksheet
explicit plot example];
1.4: 45 (if your cell phone were waterproof: "Can you hear me now?"
attentuation of signal--characteristic length),
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 [CSI problem (Newton's law of cooling):
pdf, .mw please read 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];
Optional challenge problems (more fun/interesting but safe to
ignore):
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).
- F: Quiz 2 thru 1.4; 2-4pm Mendel Award lecture by
Nobel Laureate (usually a terrific talk for people interested in science:
info,
talk info);
online 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;
rightclick "Simplify, Simplify" may be necessary to simplify the result]
> y ' = x y , y(0)=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.
If you are at all confused, try to do this way before Test 1 in week 4.
WEEK 3[-1]:
- M: paper only
class roster handout for forming
partnerships: try to partner up with a neighbor (groups of 2 or 3) at the end of class or by
Wednesday;
online handout:
how to
understand the DE machine (example of switching variables);
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 (antiderivatives
as definite integral functions, more explanation of error
function)],
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 (don't submit a Maple worksheet without consulting this
to be sure your text comments are correct).
- W:
Maple1.mw hw due any time next week, but preferably before the Th Test
(read Maple HW page);
1.5 (tank mixing problems): 37
(Use
Eq. 18 in the book; extra questions: what is the final concentration of salt?
How does it
compare to the initial concentration and the incoming concentration?);
[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] [soln]
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,
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:
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: handout on solution of logistic DEQ
[directionfield,
integral formula,
characteristic time];
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 [READ CAREFULLY: 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 M and
kM needed to use the solution formula to solve problem 16],
16 [ans: P = .95 M
after 27.69 -> 27.7 -> 28 months].
- F: Quiz 3 on linear DEs;
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)].
WEEK 4[-1]:
- M: air resistance handout;
[optional reading to show what is possible: comparison
of linear, quadratic cases; numerical solution
for any power];
2.3: 1, 2, 3,
9 [remember weight is mg, so mg = 32000 lb determines
m = 1000 in USA units, convert final speed to mph
for interpretation!],
22 [this is the tanh case: vterminal = 20.7 ft/s, t = 8 min 5 s][solution,
.mw].
- W: 2.4: 1 (do by hand and calculator, can check with euler.mw if you wish);
optional read about improved Euler;
maple2.mw:
instructions for this second maple assignment.
[due the week after Test 1 roughly; make sure you have at least one
partner and follow the submission instructions].
- Th: Test 1 thru chapter 1. [Let me know if this is a bad day because of
other tests, we can work something out.]
|Read
Test Instructions before coming to class.]
- F: 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:
Test 1 back, check answer
key; confirm grades entered correctly in Blackboard;
handout on RREF
(Reduced Row Echelon Form, section 3.3) [see bob's
first 2x2 example];
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?]
For physics majors: here is what Maple can do for
motion of a bead along
a wire. [From the
Maplesoft Application Center].
Just 2nd order force equals mass times acceleration equations, solved
numerically. We will get to this later but only for linear DEs.
- W:
maple2 is due by end of 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
bob as 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 (refers
back to system in 3.2 HW) [Matrices for doing tonight's homework with Maple
preloaded]
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 get RRED form of a matrix in Maple, right click on output region
matrix, choose Solvers and Forms, Row Echelon Form,
Reduced.
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]]); #[row by
row entry, you enter matrix using palette instead]
ReducedRowEchelonForm(A); #[you can right click reduced this: solvers and forms]
#[Cntrl SpaceBar will autocomplete menu for any Maple command you start
entering]
BackwardSubstitute(%);
To step by step reduce a matrix (max 5x6) and solve the system do (pick
Gauss-Jordan):
> LinearSolveTutor(A)
Control SpaceBar turns on command autocomplete on an input line. If
you type
> Back
and hit this key combination it will give you a menu of choices to pick
BackwardSubstitute (even I cannot remember this command past Back) which is
the key command to solve a system once an augmented matrix is completely
reduced, which you can do by a right click menu choice. "Red" will
autocomplete to ReducedRowEchelonForm.
You can bypass all of this by using the linear
solve tutor from the Tools Menu:
Maple
Tools Menu: Tutors, Linear Algebra, Linear System Solving: Gauss-Jordan elimination
- 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 or by using the line from the Maple template on
the previous day 18];
MAPLE
chemical reaction problem* [pdf] (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: can you say "homogeneous"?
[online handout only]
- F: Quiz 4;
finally matrix multiplication!;
3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
Matrix algebra is easy in Maple
[see here for how to do Matrix stuff in Maple]. <<<<<<<<<<<
WEEK 6[-1]: are you using
your VU webspace? its a shame not to play with it.
Maple2.mw is due.
- M:
handout
example for matrix inverse algorithm;
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 Standard Operations menu]
> A B [note space between
symbols to imply multiplication]
> A-1
> |A| [absolute value sign
gives determinant]
> A%T
[transpose, we don't need it]
Memorize:
.
Switch diagonal entries. Change sign off-diagonal entries. Divide by
determinant.
Always check inverse in Maple if you are not good at remembering this.
- W: read on-line handout:
determinants and area etc [why
the transpose?]
(to combine vectors we arrange them into rows of matrix,
transpose maps columns to rows, rows back to columns);
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? [this is the
final problem of maple3.mw.]
MacArthur Prize and solving linear equations with millions of terms [philly.com,
google]
- Th: now we look at linear system coefficient matrices
A as collections of columns A = < C1| ... |Cn >,
matrix multiplication on the right by a column of
coefficients as taking linear combinations of those columns,
A x = x1 C1 + ... + xn Cn
and solving homogeneous
linear systems of equations A x = 0 as looking for linear
relationships among those columns;
4.1: 1; 5, 7;
9; 15, 17; 19, 23; 25
[use technology in
both 1) evaluating the determinant and 2)
rightclick menu to the ReducedRowEchelonForm
needed
for these problems];
[HWready row reductions 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]
- F: Quiz 5;
vector spaces and subspaces;
study the handout on solving linear systems revisited
[remember this 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, 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].
WEEK 7[-1]: maple3.mw due anytime
this week thru weekend
- M: book catches up with us: linear
independence (A x = 0),
express a vector as a linear combination (A x =
b); span of a set of vectors;
handout on
linear combinations, forwards and
backwards [maple to visualize];
[optional worksheet to read: 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.
- W: bases
of vector spaces, subspaces [plane example];
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!)].
handout on nonstandard coordinates on R2 and
R3 [goal: understand the jargon and how
things work] [in class practice: .mw,
.pdf] ;
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
to hand in Thursday.
[You could also edit the Maple version
of the example to see that your hand work is correct.]
- Th: handout summarizing
linear vocabulary;
[maple
illustration of how to describe a plane in space using a subspace basis].
[optional on-line handout:
linear system vocabulary]
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 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].
- F: Quiz 6;
Fall Break. :-) Enjoy. Be safe in your travels.
WEEK 8[-1]: midterm grades Maple thru Maple1.mw; upgrades possible till
Tuesday;
spreadsheet calculation of letter grade average;
Maple3 due;
- M: Memorize: y ' = k y < -- > y = C e k x
y '' + ω2 y = 0 < -- > y =
C1 cos(ω x) + C2 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[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
or if x is not the independent variable, then use function notation:
>
y''(t)
+ 3 y'(t) + 2 y(t)
= 6 et , y(0) = 4, y'(0) = 5
- W:
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?].
- Th: 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];
[optional: read the on-line handout on
visualizing the initial value problem (IVP)
if you wish to see the interpretation of solving the IVP]
- F: Quiz 7; bob talks at 3pm in physics
colloquium (our usual
classroom):
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 [example,
DE], 4 quadrant examples:
mw,
pdf).
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; or right click on output of eqn, choose solve)
[Note: > fsolve(...) returns all numerical roots of a polynomial].
WEEK 9[-1]: maple 3 should have been turned in long ago.
- M: Hurricane Sandy lost day.
- 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].
handout on exponentially modulated sinusoidal functions [maple
videos showing different phase shifts and frequencies];
do the short problem at the end of the handout (ignore C and W) after
reading it. [soln: pdf,
mw]
Review problem session at 5:00pm today in the MLRC
- Th: Test 2 on chapters 3 and 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 this handout sheet, Maple: *check this with dsolve,
(pdf
solution; the algebra works if you are careful!)];
5.5: 1, 3, 9 [see PDF solution after you try this yourself:
version 1,
version 2]
[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].
EE majors if you are interested: RLC circuit application: RLC circuits [maple
plots]
- W: QUIZ 8! linear homogeneous constant coefficient 2nd order DEs:
complex root solutions, identifying natural decay rate/time, natural
frequency, Q factor, solve IVP
(like HW problem 5.4.14 get solution from Maple since even problem).
handout on damped harmonic oscillator driven by sinusoidal driving function
[in class example (pdf), maple resonance plots: general, specific];
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
Maple worksheet about the interpretation of
the solution.
EE majors if you are interested: driven RLC circuit problem:
R
= 16 ohms, C = 1/40 farads = 0.025 farads = 25 millifarads, L
= 8 henries, E = 4 sin(2 t) volts. Find I(t). [pdf,
mw].
Also problem 38 is equivalent to an overdamped AC voltage driven RLC circuit
problem. Check it out. [EE RLC circuit page].
- Th: 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 envelope function [and its period is 2(2
π)/(A-B)], 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 (be sure to plot one
full period of the envelope function to see 2 beats) and include only that HW problem
section in your maple5 worksheet],
11 [then convert steady state solution to phase shifted cosine at
end of
calculation], 17 [pdf],
earthquake!: 23 [pdf].
handout on beating and
resonance; [resonance
calculation side by side]
Maple plotting of multiple functioins
>
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. We need this for envelopes!
- F: Quiz 8 should be a wakeup call for some of you. Office hour visits
are the first line of defense.
Check blackboard grades; Test 2 back (check
answer key);
catch up on HW and Maple5 due any time next week (please take it seriously);
[Tacoma Narrows Bridge collapse (resonance NOT!):
engineering
explanation;
Wikipedia;
Google (choose a
You-Tube video,
lowest bidder!)];
Transition back to linear algebra:
Fun.
Watch the
MIT Eigenvector
4 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 directionfield
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. 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.
6.1: 1, 9 [1, 9 mw].
WEEK 11[-1]:
- M:
recall old handout
coupled system of DEQs [motivation: direction fields for Maple (DEPlot
directionfield) help visualize eigendirections of a 2x2 matrix]
examples: A = <<0,1>,<1,0>> [done]; <<2,0>,<1,3>>; <<1,2>,<2,4>>;
3x3 example;
6.1: 1, 9 , 13, 19; 25,
27; we will do 13 in class together;
[do everything by hand for a 2x2 matrix; for 3x3 or higher, go thru process: use Maple determinant: |A-λI|
= 0 (right click on matrix, Standard Operations), then solve to find characteristic equation and eigenvalues,
back sub and solve by rref, backsub, read off eigenvector basis, NO POLYNOMIAL DIVISION!].
- W: diagonaliization
[now revisit
coupled system of DEQs in a handout giving a preview of what we are about to embark on:
why diagonalization?];
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
it is 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>; [make a
rough drawing of the new axes and the initial data decomposition along the
eigenvector directions, or use the phaseplot worksheet to make the diagram];
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 diagonalization and matrix notation for linear systems of
DEs. [see 11S quiz 9];
handout on 1st order linear
homogeneous DE systems (complex eigenvalues) (2-d example: [phaseplot]);
handout on transition
from a complex to a real basis of a linear DE solution space [double
handout print];
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 for that
problem].
Saturday:
Jawani!
A terrific annual show with food that is lots of fun. 8 bucks. How can you
beat that?
GET TICKETS HERE:
http://bit.ly/R29unx
WEEK 11: Maple5.mw due
- M: online 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!);Thanksgiving Break.
This talk is REQUIRED WATCHING for any educated human
being:
http://www.skeptic.com/past-lectures/
[now online]
Sean Carroll
The Particle at the End of the Universe:
How the Hunt for the Higgs Boson Leads Us
to the Edge of a New World
WEEK 12:
- M:
Read 7.3 [example 2 in book, see especially
compartmental analysis example 4, which has complex
eigenvectors (more examples)];
do 7.3: 37 [closed 3 tank system with
oscillations (explanation,
text example .mw), plug into Eq(22),
solve by eigenvector method].
[solution: pdf,
mw]
- W:
summary handout on
eigenvalue decoupling so far; use
it to solve:
7.3.19 with initial conditions <x1(0),x2(0),x3(0)> = <-2,3,2>, express
solution in vector form as an explicit linear combination of the
eigenvectors, collapsing the repeated eigenvalue terms to one vector term;
7.3.26 [complex eigenvalues, check solution with maple].
Find IVP solutions for these two homework problems using Maple, then using
Maple's eigenvalues and eigenvectors, solve each problem step by step
carefully, writing down every step. Make sure your hand final answer agrees
with Maple (copy and paste!):
>
x1'(t) = 4 x1(t)+x2(t)+x3(t), x2'(t) = x1(t)+4 x2(t)+x3(t), x3'(t) = x1(t)+x2(t)+4 x3(t), x1(0) = -2, x2(0) = 3, x3(0) = 2
> x1'(t) = 3 x1(t)+0 x2(t)+x3(t), x2'(t) =9 x1(t) - x2(t)+2 x3(t), x3'(t) =
-9 x1(t)+4 x2(t) - x3(t), x1(0) = 0, x2(0) = 0, x3(0) = 17 [end number
coefficients are integers!]
MLRC 5pm volunteer test 3 problem session.
- Th: handout on extending eigenvalue
decoupling (to nonhomogeneous case, and second order);
examples handout [.mw]
(lower rightmost diagram incorrect!); go over this
handout to see how we can easily extend our present tool to cover the
nonhomogeneous case, and when we have a second derivative instead of a first
derivative;
solve x''(t) = A x(t) + F(t),
where x(t) = <x1(t),x2(t)>,
for A = <<-5/2|3/2>,<3/2|-5/2>>, F(t) = 0, x(0) = <1,0>, x'(0) = <0,1>,
make a hand plot of the eigenvectors and new coordinates y1,y2, the initial data position vector x(0) and the velocity vector x'(0) with tail at the tip of the
initial position vector;
express the mode variables y1(t), y2(t)
in phase shifted cosine form so you can compare the amplitudes and
phaseshifts of these two oscillations.
Check your solutions with Maple by solving the corresponding scalar
differential equations for x1(t), x2(t).
The following two commands will show you the two relevant plots of your
results if you are curious (replace unknown variables by their expressions
that you find):
> plot([x1(t), x2(t),t
= 0..2π]) a nice figure 8 ish design!!
> plot([x1(t), x2(t)],t
= 0..2π, color = [red,blue])
.
- F: take home Test 3 out; read the short instructions on the test and the
long instructions online;
here is a stripped down example
for the direction field and solution plot versus t, all you need for
problem 3.
WEEK 13: MLRC test 3 volunteer session day 46. Take home test 3 out for
one week day 48.
- M: in class Thursday assigned HW exercise, then do with driving
function: F(t)
= < 0,50 cos(3t) > ;
[pdf
solution of day 47 hw (undriven case)] SOLUTION!
- W:
read
handout on reduction of order with exercise
and finish exercise begun in class
[in book you can read 7.1 excluding examples 5-7 (these are the opposite of
reduction of order, not needed)];
[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.
[pdf
solution of day 48 hw (driven case)][beautiful
Maple animations, play with initial
conditions and driving frequencies!]
SOLUTION! and
the JPG movie
- Th:
2 mass 3 spring systems:
2mass-3spring-figure8curves.mw; this has the movies:
figure8curve.mw,
4 page 2 sheet handout
figure8curveset.pdf
- F: Test 3 due back unless you have extenuating circumstances and are
forced to ask for an extension. NOT!
If you feel you can do a better job by turning in the test Tuesday, you may
do so. Otherwise turn in the test today in class.
Check blackboard grades for maple, quizzes, test to be sure they are
correct.
Day 47 problem continued with simple damping using reduction of order:
x1''(t) = - c x1'(t) - 5/2 x1(t) + 3/2 x2(t),
x2''(t) = - c x2'(t) + 3/2 x1(t) - 5/2 x2(t),
x1(0) = 1, x2(0) = 0, x1'(0) = 0, x2'(0) =1;
set up 4x4 problem (DE plus inits in matrix form for <x1,x2,x3,x4>)
with c = 1, then solve with c = 1.
Use Maple to solve and plot the c = 1 solution separately and in the
phase plane to see what the behavior is.
You can add the damping terms to the solution/movies in
figure8curve.mw.
WEEK 14:
- M: Bring laptops, community service day correcting class member maple
worksheets that are still not [2/2].
Use the movie at the end of updated
figure8curve.mw to visualize what
happens when you use the same initial data but make the driving frequency
equal to 11/10 or 21/10. Use the no damping version first. then if you like
see what a damping parameter does. First execute the whole worksheet. Then
choose View, Collapse All sections get rid of the clutter, then scroll to
the bottom to see the videos.
Think of it as a video game. Then look at the movie of the moving masses as
well.
HW: make sure you resubmit all Maple files that are not already 2/2.
- T =F: remaining Test 3 returned in class;
7.4: 3,9 in class exercise with driven 3 spring,
2 mass system, easy numbers, resonance.
Finish for HW.
- W: yesterday part 1 solution stopping at IVP solution: [.pdf,
.mw];
resonance calculation;
today we part 2 "cooperative"
damping in reduced order approach to find the two natural modes and their
decay times and quasiperiods and relative amplitudes and phase shifts, using
Maple for eigenvalues and eigenvectors without worrying about initial
conditions.
HW: look at past final exams
[11F and 11S].
- Th: Past final exam play and class evaluations.
Fun: equal 2 masses and 3
springs, aperiodic motion,
periodic motion
Friday afternoon: 4pm voluntary MLRC problem session for final exam.
Sat: Final Exam 10:30-1:15, 1:30-4:00. Switching between sections allowed
with permission (email bob).
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.
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 4
Test 2: week 9
Test 3: Take home out
, in week 12
Final Exam: MLRC problem session
FINAL EXAM:
[switching between these slots with permission]
2705-02: MWF/Th 11:30/12:00 class: Saturday Dec
15 10:45 - 1:15
2705-01: MWF/Th 12:30/01:00 class: Saturday Dec
15 1:30 - 4:00
MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS
12-dec-2012 [course
homepage]
[log from last time taught][log
from penultimate time]
extras, unused stuff
- MIT linear algebra on-line videos
- Duke University on-line linear
system interactive applets
- RLC circuits for
electrical engineering and physics students
- earthquake building vibration
modeling
- the 2 spring system with damping and 2 resonance peaks:
2spring.mw
- Find general solution: x' = A x + f with
A = <<2,4>,<5,1>>, f = <3 exp(t), -t2>,
using method of undetermined coefficients on decoupled equations.
-
Optional Light Reading 1.5: Application: see how a slight generalization of
the directionfield example 1.3.3 (run
backwards right to left) to
include an initial time parameter and time and temperature scale parameters
has a useful physical application on pp.58-60. 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.
-
3 tank: pdf,
.mw
