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). It will not be collected.

- M (August 27): 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 quickly. I filter my email.**

**DURING CLASS (THIS IS THE FIRST DAY PLAY PART)**your favorite browser. (Y

On your laptop if you brought it:

1) Open*ou can 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 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, or directly:

[ http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],Maple 2018 if you already have it

3) Open

[or click on this maple file link to download the worksheet, then open it: firstdayplay.mw]

here is an example of a PDF handout: firstday2015.pdf

[here is an example of a Maple worksheet;

don't worry about page 2 or this worksheet firstday2015.mw; we will come back to it later in the course]

4)

**AFTER CLASS (THIS IS THE HOMEWORK)**to

5) log on**My Nova, choose the student tab,**and go to**BlackBoard**and look at the**Grade book f**or 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.]*

Find 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*(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.]**drop by my office St Aug 370**

**6)**Download Maple 2018 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, I will help you in my office if you wish). 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.

**7) Read computer classroom /laptop etiquette and check out the university academic integrity site. Browse our class homepage and read the linked pages. The homework problems are few so that you can get familiar with the website.**

**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).Optional: get acquainted with Maple clickable calculus DE entry and "odetest" for problems 3,5,13 [even 33!]

It is important that you read the section in the book from which homework problems have been selected before attempting them.

**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*]is inversely proportional to

y*x*:*y*∝ 1*/x*means*y*=*k/x .*is inversely proportional to the square of

y*x*:*y*∝ 1*/x*means^{2}*y*=*k/x*^{2}

- T: return your schedule forms at the beginning of
class;

check your data entered by bob on sign-up sheet---correct errors;

1:1: 7a [i.e., only check y_{1}(x)], 23 [see Maple plot (execute worksheet by clicking on the !!! icon on the toolbar)];

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;

**r**ead the paper handout: algebra/calc background sheet [online only: more rules of algebra NOT!];

on-line handout: initial data: what's the deal?

- W: 1.2 (antiderivatives as DEs): 1, 5, 15,

19 [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}(5) =*x*_{1}(5); soln: pdf, mw],

25 (like lunar landing problem, see example), 39, 40 (ans: 2.4mi), 43 (pdf) [convert to appropriate units!]. - F: course info handout;

Quiz 1 [see the archive for similar quiz 1 from the 16f semester];

1.3: 3: [hand draw in all the curves on the full page paper printout supplied by bob: projectable image]

11, 15-16 [note Df is just df/d_{y}*y*above Eq (9) in the text]; 27 [don't worry about making a diagram, just view this worksheet: .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**].**WEEK 2[-1]:**M: Labor Day

- T: Quiz one
answer key;

class directory list (paper only for privacy reasons);

**use classlist contact info**(paper only) and MyNova class photos to get to know others in your class or the other one, form Maple partnerships of 2 or 3 (4 with permission), no exceptions, you can change partners with each assignment;

1.4 (separable DEs): 1, 5; 21, 25, 27, 29

[ignorable unless you are curious: example 2: technology hint, other complications visualized]. - W:
handout on
exponential behavior/ characteristic time [cooking
roast in oven remarks]

[read this worksheet explicit plot example, execute worksheet first with !!! icon on toolbar];

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):**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.** - F: Quiz 2 (separable DEs);

online ahandout: recipe for first order linear DE [plot];

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" 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**WEEK 3[-1]:** - M: online handout:
how to
understand the DE machine (example of switching variables);

online discussion: antiderivative functions defined by definite integrals;

1.5: 17, 26 [ans:*x*= ½*y*^{ -2}+C*y*^{ -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: Δ*t*in years is the fraction of the year, so S(*t*) Δ*t*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 Δ*t*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*) Δ*t*; adding these and dividing by Δ*t*, taking limit to get dA/d*t*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 the hand written solution retirement.pdf (don't submit a Maple worksheet without consulting this to be sure your text comments are correct).

(read Maple HW page).*Maple1.mw hw due any time after This Wednesday through Thursday next week, but preferably be familiar with using Maple to check your work before the next quiz*

- T:
1.5 (tank mixing problems ChE majors especially):

37 (Use Eq. 18 in the book or the boxed equation in the online handout);**extra questions to answer: what is the final concentration of salt? How does it compare to the initial concentration and the incoming concentration?**); [solution pdf, mw]

[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**(as unknown),*y***linear in**(as unknown), some combination of these three, or*x***NOTA**(none of the above, i.e., from skipped section 1.6), for example d*y*/d*x*=*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. - W: 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*B*_{o}=*kMP*_{o}and*D*_{o}=*kP*_{o}^{2}, and these are easily solved for*M*and*kM*in terms of*B*_{o},*D*_{o}, and*P*_{o}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 (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, rightclick choose RHS, then rightclick choose Plots, Plotbuilder, choose horizontal window based on characteristic time);

plot this solution with technology and choose a horizontal window (negative and positive*t*values) in which you see the**reversed**S-curve nicely,the approximate time interval for an appropriate viewing window*be ready to give***in class**

11 ["inversely propto sqrt": use β =*k*_{1}/*P*^(1/2), δ =*k*_{2}/*P*^(1/2) in eq.(1)],

23 [Note d*x*/d*t*= 0.004*x*(200-*x*) so*k*= 0.004,*M*=200 for logistic curve solution formula],

30, 31 [ans: find β_{o}= 0.3 from condition d*P*/d*t*(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)].**4pm Villanova Room One Book Villanova and Peace Award Talk. Don't miss it!**

[It was the best one I have attended in many years, see a brief 24 minute version here.]

**WEEK 4[-1]: maple1.mw due anytime through approximately Friday (so you are ready for using it on Test 1)** - M:
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, 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:*v*= 20.7 ft/s,_{terminal}*t*= 8 min 5 s].[soln, .mw] - T:
2.4: 1 (do by hand and calculator, can check with euler.mw if you wish:
soln);

**optional:**read about improved Euler in textbook;

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]. - W:
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.

- F: Quiz 4;

handout on RREF (Reduced Row Echelon Form, section 3.3) [see bob's example 3 using Maple];

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) ;

[Matrices for doing some of tonight's homework with Maple preloaded];

youlearn a technology method since this is insane to do by hand after the first few simple examples);**must**

23 [can your calculator handle this?]

You can get to the Linear Solve Tutor from the Tools Menu:

**Maple Tools Menu: Tutors, Linear Algebra, Linear System Solving: Gauss-Jordan elimination**

To just get the result instead:**Click**on augmented matrix entered with the Matrix palette, then from the right pane menu choose**Solvers and Forms**, then**Row Echelon Form**, then**Reduced**to get the RREF form of the matrix, then from the right pane menu choose**Solvers and Forms**, then**Linear Solve**, and you get the RHS of the solution with no LHS variables assigned to these expressions because Maple has no clue what variable names you wish to use.

See Example 3: row ops with MAPLE.

**WEEK 5[-1]:** - M:
**Bring laptops to class for hands on row reduction tutor experience;**

handout on solving linear systems example [.mw];**we always want to do a full [Gauss-Jordan] 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 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 final result, compare with the right click solution].

5:30 voluntary test 1 problem session in MLRC. Come ask any last minute questions. - T:
**Test 1**on 1st order DEs; come early for first class, stay later in last class if you need a bit more time.

- W: 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 right pane RREF of the augmented matrix, then right pane Linear Solve;

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 (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] - F: no quiz (test week);

finally matrix multiplication! [handout];

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]:** - M:
**Test 1**back, check out answer key; check BlackBoard grades;*"I believe in second chances, I drop the lowest of the first two test grades if the successive test grade is higher."*

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, as in the alternative derivation of the matrix inverse];

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, or just right click*]

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.

- T: optional on-line handout:
determinants and area etc ;

[optional: why the transpose? (tovectors we arrange them into rows of matrix, the transpose maps columns to rows, rows back to columns)];**combine**

Read explanation of why we need determinants;

3.6: (determinants abbreviated:to understand)**forget about minors, cofactors, only need row reduction evaluation**

13, 17, 21 (plug in just so you've done it once);

17* [use this example to 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? [test to see if you are paying attention]**

[this is the final problem of maple3.mw. make sure you find a partner or join a 2 partner group for this assignment]

- W:
**linear independence of a set of vectors;**[motivating example]now we look at linear system coefficient matrices A as collections of columns**A = < C**matrix multiplication on the right by a column of coefficients as taking linear combinations of those columns,_{1}| ... |C_{n}>,

A x = x_{1 }C_{1}+ ... + x_{n }C_{n}

**A x = 0**as**looking for linear relationships among those columns (nonzero solns=linear rels) ;**

4.1: 1; 5, 7; 9; 15, 17; 19, 23; 25 [use technology in both 1) evaluating the determinant and 2)*ReducedRowEchelonForm*needed for these problems];

[HW ready 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 [like 16F Q4];

**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 vectorsresulting from the solution of a linear homogeneous condition*x***A**=*x***0**can be aof a vector space; interpretation: lines, planes, ..., hyperplanes through the origin];**subspace**

4.2: 1, 3, 11; 15, 19.**Try to finish up Maple2 this weekend. Try to submit Maple 3 by the end of next week if you can.**

WEEK 7[-1]: - M: 4.3: book catches up with us:
(

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 colmns of?];*A*of a set of vectors; handout on linear combinations, forwards and backwards [maple to visualize];

span

[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.

- T:
**bases of vector spaces, subspaces;**[counterexample]

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 R^{2}and R^{3}[goal: understand the jargon and how things work]

[in class practice on blank grid [ .mw, .pdf] with*b*_{1}= <3,-1>,*b*_{2}= <2,2> ; what are new coords of <-2,6>? what point has new coords <2,-1>?] ;

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 corresponding blank worksheet handed out in class for a new basis transformation matrix and its inverse given on that sheet, 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. [Let the larger grid squares of a centimeter determine your unit tickmarks!]

Then confirm your matrix calculations graphically by following the completed example: 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 and confirm that this agrees with your previous calculation. 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 your calculations.**Put your name on it and bring it to class to hand in next class**. [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.]

- W:
hand in worksheet;

handout summarizing linear vocabulary for sets of vectors;

[optional on-line 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: 15 [Hint: solve c_{1}(1 +*x*) + c_{2}(1 -*x*) + c_{3}(1 -*x*) = 0, for unknowns c^{2}_{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],

16, 17 [same approach].

check out the Practice Quiz 6 [with answer key after you try it yourself] before Friday.

HW worksheet solution. - F: Quiz 6 on new coordinate grids in the plane.

**FALL BREAK**:-) Enjoy. Be safe in your travels.

**WEEK 8[-1]:** - M:
**Memorize:***y ' = k y*< -- >*y = C e*^{ k x}

*y '' + ω*= 0 < -- >^{2}y*y = C*cos_{1 }*(ω x) + C*sin_{2}*(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"]

- T: handout on sinusoidal example;
repeated root plot;

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*]); (let*a*= 0 here to plot only*x*> 0)

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?]. - W: online handout:
wronskian and higher order constant coefficient linear homogeneous DEs;

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 [see 16F quiz 6 but you find the solution basis functions
yourself];

complex roots: handouts on 1) the amplitude and phase shift of sinusoidal functions,

2) 4 quadrant amplitude-phase shift examples: [mw, pdf].

and 3) DE example [.mw];

5.3: 8 [ans: y = exp(3*x*) (c1 cos(2*x*) + c2 sin(2*x*))] , 9; 17; 22,

23 (express in phase-shifted cosine form, see first handout).**You should turn in Maple 3 by Monday**(be ready to use Maple on Test 2!),

and should have started on Maple 5.**WEEK 9[-1]: quiz 7 makeup, ask bob; pick up quiz 6 if absent Friday** - M:
**MLRC voluntary problem session**at 5:30pm; no office hour 12:30-1:45;

T: 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 (or just right context menu to get the solution), andthe IC linear system solution as in the template, edit away all stuff not directly relevant to your new problem];*also check*

handout on exponentially modulated sinusoidal functions [EEs read the complex exponential stuff]

[maple videos showing different phase shifts and frequencies];

do the short problem at the end of the handout (ignore complex C and W unless you are EE or physics oriented) after reading it. [soln: pdf, mw].

- T: Test 2 on chapters 3,4.
- W: handout on linear homogeneous 2nd order DEQ with constant positive coefficients
(damped harmonic oscillator)

[on-line only: examples in nature];

5.4: 1, 3 [use meter units!],

13, 14 [b) the solution once put in the form*x*= A exp(-K*t*) cos(ω*t*- δ) has the "envelope curves"*x*= ± A exp(-K*t*) ] ,

17; 23 [pdf; this is a useful application problem, please study the solution carefullytrying to solve it yourself];**after**

We will work on 17 in class to reproduce the Maple results. - F:
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,

(consult after doing it by hand first: 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],

33, 38 [*y*= ( exp(-*x*)(176 cos(*x*)+197 sin(*x*)) )/85 - (6 cos(3*x*)+7 sin(3*x*))/85]; [not many of the book 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].

**ignore this: EE/Physics majors**if you are interested: RLC circuit application: RLC circuits [maple plots]

**WEEK 10 [-1]: check blackboard grades; answer key for quizzes, test 2** - M: 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: 8. Ignore the book instructions and solve this problem by hand for the initial conditions*x*(0) = 0 =*x*'(0) (at rest at equilibrium).

[Read the Maple worksheet about the interpretation of the solution. Wiki: driven harmonic motion, amplitude plots: they introduce ζ = 1/(2Q)]

15, 17. [Follow the handout procedure as in this particular case, find the amplitude function of the frequency and plot it to compare with the back of the book. Evaluate the natural frequency and*Q*value for each.

solns: 8: [.mw, .pdf]; 15, 17: [.mw, .pdf] - T: 5.6:
handout on beating and
resonance [with HW explanation];

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*T*= 2(2 π)/(A-B)], look at the plot of x and +/- this amplitude function (the envelope) together in this beating example worksheet, then do:

5.6.1*: as explained here,**repeat**for the HW problem described in the above 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) 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];

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]

**Maple plotting of multiple functions**

>*x, -x, x*sin(*x*)^{x}(2 cos(*x*) +sin(*x*)) you need a space before the left parenthesis!]

If you are curious watch the 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 (You-Tube video)];

a real resonance bridge problem occurred more recently: the Millenium Bridge resonance.

Line up to VOTE! - W:
**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]

matrix examples:*A*= <<0,1>,<1,0>> [done]; <<2,0>,<1,3>>; <<1,2>,<2,4>>];

watch the MIT Eigenvector 4 minute video [there are 6 frames which then repeat, so**stop**when you see it beginning again---from 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: 1,2,9. 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, 9 [1, 9 mw]. Ignore polynomial division discussion, use technology for roots of polynomials!

- F: Quiz 8 (like 16F quiz 7);

Maple 3x3 matrix eigenvector example handout [.pdf, .mw]; more examples;

6.1: 13, 19; 21, 25 (upper triangular so diagonal values are eigenvalues!),

[Maple input for eigenvalue derivation];

27 (complex!);

[do everything by hand for 2x2 matrices;

for 3x3 or higher, go thru process: use Maple(right click on matrix, Standard Operations), then*determinant*: |A-λI| = 0*solve***(A-λI) 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].

[repeated eigenvalues: <<9, -6, 6> | <4, -1, 4> | <0, 0, 3>>]**WEEK 11[-1]:** - M: 6.2: diagonalization [now revisit
coupled system of DEQs in a handout giving a
of what we are about to embark on:**preview****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**= Band**y****y***= B*to find the new coordinates of the point <^{-1 }**x***x*_{1},*x*_{2>}= <-2,4> in the plane and to find the point whose new coordinates are <*y*_{1},*y*_{2>}= <2,-1>; finally evaluate the matrix product by hand*A*to see that it is diagonal and has the corresponding eigenvalues in order along the diagonal.[multiply_{B}= B^{-1}AB**b1**and**b2**by*A*to see what multiple you get in each case---the eigenvalue];

[solution hint: 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" and what kind of special matrix is this if Delta is zero?]. - T:
handout on the geometry of diagonalization
and first order linear
homogeneous DE systems

(2-d example: real eigenvalues [phaseplot]), this is the solution of yesterday's diagonalization problem)

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 systemfor the matrix of problem 6.1.19, with initial condition**x' =**A**x****x**

3) find the eigenvalues and matrix*B*of eigenvectors for the matrix*A***= <<**0**|**4**>**,**<**-4**|**0**>>**(input by rows) and verify that*A*is diagonal._{D}= B^{-1}A B

- W:
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 [4 page double handout print];

Find the general solution for the DE system*x*' =*A**x**A***= <<**0**|**4**>**,**<**-4**|**0**>>**(input by rows) and then the solution satisfying the initial conditions**x**

Repeat the process for problem*x*' =*A**x**A***= <<**1**|**-2**>**,**<**2**|**1**>>,**initial conditions**x**

[if you like you can quickly edit the DEPlot phaseplot template to reproduce the back of the book diagram for that problem].

Th-Sat VST All My Sons - F: Quiz 9; Read 7.3
**optional**read with many extras, only for the curious:

[**compartmental analysis: real eigenvalues:**example 2 (open case+comparison with closed case: real/complex eigenvalues)**,**: example 4 (closed case only); (more examples 7.3.35,36)];

complex eigenvalues

do 7.3: 37 [closed 3 tank system with oscillations (explanation, text example4.mw), plug into Eq(22), solve by eigenvector method] (solution on-line: .mw, .pdf);

choose one problem from 7.3: 27-30 (solution on-line: .mw).**for complex eigenvector matrices**that do not come out in "rationalized" form but involve a product or quotient of complex numbers, right context on matrix, select "**map command onto**" and insert the command "**evalc**" to evaluate to a complex number, expressing each entry as its real part plus*i*times its imaginary part. [or see this worksheet].

Nov 16 2pm Mendel Award Talk [background][arrive by 1:30 to be sure to get a seat]

Pharmacokinetics: multicompartmental analysis of drug passage in the body is like a multitank problem.

**WEEK 12 [+1]:** - M:
summary handout on
eigenvalue decoupling so far;

in class open and execute this Maple worksheet: eigenvector DE solution workshop:

x*A**x*

7.3: 31 (be sure to maximize the third variable, exactly using Maple then numerically, make sure your plot of the 3 variables in a "decay window" agrees), 37, 39. - T:
handout on extending eigenvalue
decoupling

(to nonhomogeneous case, and second order, ignore bottom half for now);

online examples handout [.mw];

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*) = <*x*_{1}(*t*),*x*_{2}(*t*)>,

for*A*= <<-5/2|3/2>,<3/2|-5/2>>,(t) =**F****0**,(0) = <1,0>,*x*(0) = <0,1>, make a hand plot of the eigenvectors and new coordinates*x'**y*_{1},*y*_{2}, the initial data position vector(0) and the velocity vector*x*(0) with tail at the tip of the initial position vector; express the mode variables*x'**y*_{1}(t),*y*_{2}(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*x*_{1}(*t*),*x*_{2}(*t*).

T-Day break**WEEK 12[+1]:** - M: in class
*together*(**F***t*) = < 0,50 cos(3*t*) > ;

[interpretation coming soon---this represents an equivalent 2 mass 3 spring system: beautiful Maple animations, and the JPG movie.]

MLRC test 3 voluntary problem session 5:30. - T: Take home
Test 3 will be linked here after
class; we will start it in class;

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 4. - W: Continue working on Test 3 in class.

- F: Last day to continue working on Test 3 in class.
**WEEK 13 [+1]:** - M:
7.5:
2 mass spring systems: theory plus
worked examples [figure8curve.mw];

[solution of days 45, 46 exercises; description parts a) thru k): drivenDEsystemExercise.pdf]

plots: figure8curvemodesplot.mw (results only pdf print: figure8curvemodesplot.pdf );

HW: vertically hung with gravity? do parts a) -d).

[insane, see this calculation for the eigenvalues;**challenge**for when you have more time: show the slower mode eigenvector has same signed components, while the faster mode eigenvector has oppositely signed components, using this crazy approach; removing the third spring does make things easier! you can more easily get manageable expressions for the eigenvectors]

[2 mass 3 spring system: beautiful Maple animations, and the JPG movie] - T:
7.5: 3, 9 [same system
s previous day but different numbers, with/without driving term, same procedure as above,
namely:
driven 3 spring,
2 mass system, easy numbers, resonance (do parts a-f; wait for parts
g,h)].
- W: Takehome Test 3 back? [You may request an extension by email if you
feel more time would allow you to do a better job. Tell me your expected
submission date.] Continue working on yesterday's problem.

OFFICE HOURS POSTPONED TODAY till at least 3pm for dept job hiring meeting. Let me know if you want to stop by. - F:
handout on
day 50 example used to do a
mass spring system resonance
calculation [slider
exploration];

last concept: handout on reduction of order with exercise

[optional: mixed order DE system example: pdf, mw]

[in book you can read 7.1 First Order Systems, examples 3,4 (examples 5-7 are the opposite of reduction of order, ignore)]; homework consists of very short problems,, just rewrites, to make sure you understand matrix notation:**no solving required**

7.1: 1, 2, 8 [first let= [**x***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*

[solutions for fun??]

7.2: 5, 9.

Days 51, 52 7.5.3,9 exercise solution: [pdf, mw]; resonance analysis: pdf

**WEEK 14[-1]:** - M:
YouTube videos on damped oscillators
[tuned mass dampers!? short exercise];

reduction of order handout: scalar exercise soln, vector exercise without reduction [soln], [soln with reduction].

- T[F]: Earthquake!?;

reduction of order version of 7.5.3_9;

archived exam exercise. - W:
**Deadline for extended period return of Test 3 in class;**

CATS teaching evaluation first 20 minutes of class;

Final Exam discussion [archived exams].

Th: MLRC voluntary problem session 5:30pm??

Maybe again next Wednesday 5:30pm?? for those taking Thursday exam?

Sa: 04 exam 10:45br, Th: 05 exam 2:30

**Weeks 2 and 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 4-5**

**
Test 2: week 9**

**Test 3: **Take home out-in **week 12**

**Final Exam: **MLRC problem session Friday May 1, 3pm

**
FINAL EXAM:
[switching between these slots with permission]**

2705-04 (10:30class): Sat, Dec 15 10:45 - 1:15

2705-05 (11:30class): Thur, Dec 20 2:30 - 5:00

MAPLE and G.Calc. CHECKING ALLOWED FOR QUIZZES, EXAMS