MAT2705 14F 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).

  1. M (August 25): GETTING STARTED STUFF. By Wednesday, August 27, 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 17 or 18 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 18 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 can always access and open Maple 18 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 clickable calculus 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
     
  2. 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?
     
  3. 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!].

    Read the course info handout. Office hours are set.
     
  4. F: Quiz 1 through 1.1 (see quiz archive  for an idea of what to expect);
    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, 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]:
    Labor Day Monday??, no classes. probably no hard labor either.
     
  5. W: 1.4:  1.4 ( separable DEs ) 1, 5; 21, 25, 27, 29
    [ignorable unless you are curious: example 2: technology hint,  other complications visualized].

    Check quiz 1 answer key: Quiz 1 [answer key] [Maple check].
    If time: Right click menu driven Maple intro for DEs explained step by step in class.

  6. Th: handout on exponential behavior/ characteristic time [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];
     
     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 problem 64 (ChmE: ans r = 1/35 in). <<<<< ChE majors!
     

  7. F: Quiz 2 [you must use Maple to check your separable DE IVP solution, see archived quiz or: check your IVP solution];
    paper only handout for classlist contact info (no online PDF for privacy), look for Maple partners in either section (check photo roster in MyNova);
    handout: recipe for first order linear DE;
    1.5: 3, 7, 11, 21; 27;
    21*[check both the general solution and the initial value problem solution with the dsolve template].

    For future reference:
    > deq := y ' = x y      [space implies multiplication]
    > sol:=dsolve(deq, y(x))
    > solinit := dsolve({deq, y(0)=1}, y(x))
    [or enter DE plus IC separated by a comma, right click on output, choose Solve DE, for y(x), or Solve DE interactively;
    rightclick "Simplify, Simplify" or "Simplify, Symbolic" (with radicals) may be necessary to simplify the result    ]
    > y ' = x y , y(0)=1

    Quiz 2 answer key online now.

    WEEK 3[-1]: Weeks 3 thru 4: come by and find me in my office, tell me how things are going. This is a required visit. Only takes 5 minutes or less.
    If you are at all confused, try to do this way before Test 1 in week 4.
  8. M: 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 the hand written solution retirement.pdf (don't submit a Maple worksheet without consulting this to be sure your text comments are correct).


  9. 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: pdf, .mw]
    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    .

  10. 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].


  11. F: Quiz 3 on linear solution algorithm;
    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: Maple1.mw due any time this week through the weekend. [you must have a partner!]
  12. M: air resistance handout (example of a piecewise defined 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: vterminal = 20.7 ft/s, t = 8 min 5 s][soln, .mw].

    Quiz 3 answer key online now.

  13. W: bring laptops to class today;
    2.4: 1 (do by hand and calculator, can check with euler.mw if you wish: soln);
    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].

     read this handout on DE's that don't involve the ind var explicitly.

  14. Th: Test 1 thru chapter 1 (separable, linear DEs, checking solutions, exponential behavior: characteristic constant);
    [Read Test Instructions before coming to class.]

     read this handout on DE's that don't involve the ind var explicitly;

  15. 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]:  maple2 is due by end of this weekend;
  16. M: handout on RREF (Reduced Row Echelon Form, section 3.3) [see bob's first 2x2 example,example 3];
    3.2:     1, 3, 7; 11, 13, 15 (do a few by hand, then you may use step-by-step row ops with MAPLE or a calculator for the rest) ;
    [Matrices for doing some of tonight's homework with Maple preloaded];
    you must learn a technology method since this is insane to do by hand after the first few simple examples);
    23 [can your calculator handle this?]

  17. W: Bring laptops to class for row reduction tutor experience;
    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; then check with ReducedRowEchelonForm or by right-clicking and choosing from the menu (solvers and forms)].

    To get RREF 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.

    Now:
    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

    If we only want to solve the system, right click on the output region matrix and select Solvers and Forms, Linear Solve.
    [You can row reduce first by right click,  Solvers and Forms, Row Echelon Form, Reduced.]
  18. 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 right click RREF of the augmented matrix, then right click 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: can you say "homogeneous"? [online handout only]


  19. 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]: still not too late to come visit bob in his office! [especially those who did not do well on test 1]
  20. 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.

  21. W: read on-line handout: determinants and area etc ;
    [why the transpose? (to combine vectors we arrange them into rows of matrix, the 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. make sure you find a partner or join a 2 partner group for this assignment]

  22. 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 (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) 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].

  23. F: Quiz 5;
    vector spaces and subspaces    ;
    study the handout on solving linear systems revisited [remember the original one: solving linear systems example]    ;
     [note only a set of vectors x resulting from the solution of a linear homogeneous condition A x = 0 can be a subspace of a vector space; interpretation: lines, planes, ..., hyperplanes through the origin];
    4.2: 1, 3, 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
  24. 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.

  25. W: bases of vector spaces, subspaces [if time: pdf,  calculate  and visualize];
    4.4: 1, 3, 5, 7;
    9 [this is already reduced, find solution as in class example, pull apart to find coefficient vectors of parameters, repeat for next 4 problems reducing first],
    13; 15, 17, 25 [use technology for all HW row reductions and determinants (but 2x2 dets are easy by hand!)].

    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 corresponding blank sheet 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 following the completed 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.
    [If you are feeling ambitious,you could also edit the Maple version of the example to see that your hand work is correct.]

  26. Th: 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 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].

  27. F: Quiz 6;

    Fall Break.   :-) Enjoy. Be safe in your travels. 

    WEEK 8[-1]: midterm grades Maple thru Maple2.mw; upgrades possible till Tuesday; spreadsheet calculation of letter grade average;
    late quiz 6 takers see me today? don't forget to submit Maple3.mw
  28. 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;
  29. 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?].

    What is the key math in this movie? Solving 2nd order differential equations for light paths in curved spacetime: http://www.wired.com/2014/10/astrophysics-interstellar-black-hole/

  30. 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]


  31. F: Quiz 7;
    complex roots: handout on the amplitude and phase shift of sinusoidal functions and DE example [.mw];
          extras: [alternative discussion same example: .mw, pdf], 4 quadrant amplitude-phase shift examples: [mw, pdf].
    5.3: 8 [ans: y =  exp(3x) (c1 cos(2x) + c2 sin(2x))] , 9; 17; 22,
    23 (express in phase-shifted cosine form, see handout).

    WEEK 9[-1]: Test 2 this week, Thursday
  32. M: 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 unless you are EE or physics oriented) after reading it. [soln: pdf, mw].

  33. W:  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] (this is a useful application problem, please study the solution carefully after trying to solve it yourself);
    handout on     linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillator)
    [on-line only: examples in nature].

    Wednesday 5:30pm MLRC voluntary problem session for me to answer any last minute questions about material from chapters 3,4, archived tests.

  34. Th: Test 2 on chapters 3,4.

  35. F:  2pm C-Center V-room: Mendel Medal Lecture, featuring world-renowned epidemiologist and “microbe hunter” W. Ian Lipkin, MD: “Of Microbes and Man: A Delicate Balance.”
    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/Physics majors if you are interested: RLC circuit application: RLC circuits [maple plots]

    WEEK 10[-1]:
  36. M: catchup day. in class: 5.4,5.5 HW discussion. After class, go back and do some of these problems. Try 5.5.10 if you have done them.
    For second order DEs, if a RHS driving function term satisfies the homogeneous DE, multiply the corresponding trial function (the most general solution of the homogeneous DE for which that term is a particular solution) by the independent variable.

    Note the 5.5.9 solution now posted above and if you are curious: a slight change to this problem to make it physically interesting. handouts/ep3_5-5-9new.pdf describing an overdamped spring system at rest to which an impulse force is applied, from which one can calculate the maximum displacement which occurs.

  37. W: 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];
    5.5: 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. For the curious, Wiki: driven harmonic motion, amplitude plots: they introduce     ζ = 1/(2Q).


  38. Th: Test 2 back, check out answer key;
    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*: repeat for the HW problem at the end of this 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], no formulas needed for resonance calculation: 17 [pdf], earthquake!: 23 [pdf].
    handout on beating and resonance [includes resonance calculation for previous day example];

    Maple plotting of multiple functions
    >     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!

  39. F:  Quiz 8: linear homogeneous constant coefficient 2nd order DEs: complex root solutions, identifying natural decay rate/time, natural frequency and period, Q factor,  solve IVP (like HW problem 5.4.14).  HW: ;
     catch up on Maple5.mw homework due any time next week.

    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!): engineering explanation; Wikipedia; Google (choose a You-Tube video, lowest bidder!)].

    WEEK 11[-1]: Maple5 due any time this week;
  40. M: 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: 1,2,9. See if you can guess the eigenvector directions and the corresponding eigenvalues (all integer triangles locating the vectors and integer eigenvalues); write down a simple representative eigenvector (with the smallest integer components, say) and its eigenvalue that you can read off from the applet as explained within the introductory webpage to compare with your matrix calculations for these textbook HW problems .

    6.1: 1, 2, 9 [1, 9 mw].  Ignore polynomial division discussion, use technology for roots of polynomials!

    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>>.

  41. W: eigenvector analysis: 3x3 example (repeated eigenvalues:  <<9, -6, 6> | <4, -1, 4> | <0, 0, 3>>);
    we begin HW in class:
    6.1:     13, 19; 21, 25 (upper triangular so diagonal values are eigenvalues!),
    27 (complex!); we will do at least 13 and 21 in class together (?);
    [do everything by hand for 2x2 matrices; 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 its solutions, the eigenvalues, back sub them into the matrix equations (A-λI) x = 0 and solve by rref, backsub, read off eigenvector basis of soln space, NO POLYNOMIAL DIVISION!].

  42. Th: diagonalization [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-1 x 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"?].

  43. F: Quiz 9 [last quiz!]
    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.

    WEEK 12[-1]:
  44. M: 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 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, see pdf].

    Fact: eigenvectors for distinct eigenvalues MUST BE LINEARLY INDEPENDENT.
    So if you get n distinct real eigenvalues for an nxn matrix, you must get a basis of real eigenvectors from it.

  45. W: Quiz 10 NOT! (complete for HW);
    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.


  46. Th:  Read 7.3
    optional read with many extras, only for the curious:
    [compartmental analysis: real eigenvalues: example 2 (open case+comparison with closed case = complex eigenvalues) ,
     complex eigenvalues    : example 4 (closed case);(more examples 7.3.35,36)];
    do 7.3: 37 [closed 3 tank system with oscillations (explanation, text example4.mw), plug into Eq(22), solve by eigenvector method].

    for complex eigenvector matrices that do not come out in rationalized form, right click on matrix, select "map command onto" and insert the command "evalc" to evaluate to a complex number, meaning as real plus i times imaginary part.

  47. F:  summary handout on eigenvalue decoupling so far;
    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]) .

    WEEK 13:
  48. M: delay till post T-day:
    day 47 HW exercise, repeat together in class with driving function: F(t) = < 0,50 cos(3t) > ;
    [equivalent 2 mass 3 spring system: beautiful Maple animations, play with initial conditions and driving frequencies!  and the JPG movie].

    [see archived 13F Test 3]
    MLRC problem session 5:30pm. Take home Test 3 out in paper at session, online; budget your time, don't leave it till last minute.
     take home Test 3 will be linked here; please read the short instructions on the test and the long instructions online; this is not a collaborative effort;
    here is a stripped down example for the direction field and solution plot versus t, all you need for problem 3.

    Thanksgiving Break.
  49. M: Continue/Start working on Test 3 in class and for HW.

  50. W: Bring laptops for in class group exercise in driven second order DE systems. Work in small groups, taking each step seriously. Ask bob when you are confused.
    Check BlackBoard grades for accuracy, Maple total 7/7.

  51. Th: Bring laptop to class to continue working on Test 3.


  52. F: Test 3 due Friday or Monday in class, extension may be requested by email if Monday is not adequate.
    finish in class exercise started Day 50;
    2 mass spring systems: theory plus worked examples (read section 7.4 in textbook or bob's handout below);
    [jpg movie of 2 mass positions versus time, corresponding phase plane plot of the 3 independent modes: 
        2 natural modes plus a response mode to a driving frequency]

    handout for Monday on:
    2 mass spring systems: theory plus worked examples; 
       [in class exercise: drivenDEsystemExercise.pdf, solution: figure8curve12.pdf [included in previous line PDF]
         plot: figure8curvemodesplot.mw, figure8curvemodesplot.pdf ]

    WEEK 14:
  53. M: hand in Test 3 or email request for extension before class;
    the rest of this week is not on final exam (based on exercise from days 50,52), but to complete the ideas of the course:
    handout on     reduction of order with exercise and finish exercise begun in class for HW
    [worked examples: single variable second order ex: pdf, mw; two variable second order ex: mw, handouts/redorderex-short.mw];
    repeat reduction of order setup for driven second order system exercise, use Maple to solve first order scalar DEs with initial conditions all zero for the variables and their derivatives; 
    [in book you can read 7.1 First Order Systems, examples 3,4 (examples 5-7 are the opposite of reduction of order, ignore)].

  54. T[F]: We derive by hand in class the solution to the decoupled 2nd order equations [pdf exercise]. Solution: [pdf, mw].
    Check out the archived final exam from 13F (next day link).

  55. W: archived final exam.

  56. Th: CATS evaluations;
    MLRC 3pm MLRC voluntary problem session for final exam on Saturday.

    FINAL EXAM:  [switching between these slots with permission] room change to next door: JB202A!
         2705-02: MWF/Th  11:30/1:00 class: Sat, Dec 13 10:45 - 1:15
         2705-01: MWF/Th  12:30/12:00 class: Sat, Dec 13 1:30 - 4:00

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: Sat, Dec 13 10:45 - 1:15
     2705-01: MWF/Th  12:30/01:00 class: Sat, Dec 13 1:30 - 4:00

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

19-oct-2014 [course homepage] [log from last time taught]

extras, unused stuff