MAT2705 07F homework and daily class log

Your homework will appear here each day as it is assigned, with occasional links to some MAPLE worksheets when helpful to illustrate some points where technology can be useful. [There are 56 class days in the semester, numbered consecutively below and  labeled by the (first initial of the) day of the week.] Usually it will be summarized on the white board in class, but if not, it is your responsibility to check it here. You are responsible for any hyperlinked material here as well as requesting any handouts or returned tests or quizzes from classes you missed. Homework is understood to be done by the  next class meeting (unless that class is a test, in which case the homework is due the following class meeting).

1. M: 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-XX]", where XX is your section number 01 or 06, telling about your last math courses, your comfort level with graphing calculators (can you do symbolic derivatives and integrals on your graphing calculator if not in Maple?) and computers and math itself, how much experience you have with MAPLE (and Mathcad if appropriate) so far, why you chose your major, etc. Tell me what your previous math course was named (for sophomores: Mat2500 = Calc 3 or Mat1505 = Calc 2).
   [In ALL email to me, include the string "mat2705" somewhere in the subject heading if you want me to read it. I filter my email.]
   
   In class:
   1) log on to your computer and open Internet Explorer. (IE allows you to open MAPLE files linked to web pages automatically if MAPLE is already open or if it is available through the Start Menu Program listing under Math Applications, in Netscape you must save the file locally and then open in it MAPLE using the File Open task.)
   2) log on to 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 My Courses classroom site, and visit the link to my course homepage from it
   [ http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
   3) Open Maple 11 Standard (red not yellow icon) from the Start Menu Program listing under Math Applications
   [or click on this maple file link: Maple example file]
   [Using the File OPEN URL command in Citrix Maple, with this URL (right click, copy shortcut) will show you a Maple file of how to pick the viewing window for realistic exponential and sinusoidal example functions. Clicking on the !!! icon on the tool bar with execute this worksheet and restore the output which has been removed to make the file smaller. It will also automatically open the two subsections, which can be closed by clicking on the triangle at the section title. Try this at home to make sure you can access citrix Maple.]
   
   4) bob will quickly show you the computer environment supporting our class. And chat up a bit the course. [cell phones on vibrate or off: bob will give you his cell phone number in class, not on the web!]
   
   Afterclass:
   5) log on to MyWebCT and look at the Grade book: you will find all your Quiz, Test and Maple grades here during the semester.
   [This is the only part of WebCT we will use this semester.]
   Check out the on-line links describing aspects of the course (no need yet to look at the MAPLE stuff). Fill out your paper schedule form (get  a copy in class to fill out or print it out back-to-back earlier to fill out in advance and bring to your first class already done; consult the dorm abbreviation list) to return in class Tuesday.
   [You can drop by my office St Aug 370 (third floor, Mendel side, by side stairwell) to talk with me about the course if you wish and to see where you can find me in the future when you need to.]
   6) Make sure you can access and open Maple 11 Standard in citrixweb on your laptop. If you have any trouble, email me with an explanation of the errors. If I receive no email, it means that you were successful. The first time you use citrix on a computer, you must download and install the client from the link on that web page.
   You are expected to be able to use citrix Maple on your laptop or Windows Maple in class when needed. We will develop the experience as we go.
   
   7) Read computer classroom etiquette. Then read the first paper handout: algebra/calc background sheet.
   
   8) Homework Problems: 1.1: 3, 5, 13, 33 (this links to a PDF scan of the HW problems from the book if you have not yet purchased it);
   [ "A is proportional to B" means "A = k B" where k is some constant, independent of A and B]
   (short list so you can check out our class website and read about the course rules, advice, bob FAQ, etc, respond with your email).
   It is important that you read the section in the book from which homework problems have been selected before attempting them.
   Get acquainted with Maple11 Standard DE entry and "odetest" for problems 3,5,13 [even 33!]
   [use citrixweb Maple on your own computers]
    
2. T: return your schedule forms at the beginning of class [did you use the 3 letter dorm abbreviations?];
   1:1: 7a [i.e., only check y₁(x)], 23;
   formulating DEs: 27, 29 [Hint: recall perp lines have slopes which are negative reciprocals; make a diagram of the given point (0,1), a "generic" function graph curve, and a point (x,y) on the curve and the connecting line segment between points perpendicular to the tangent line, then compute the slope from two points, and from the derivative, equate the two to get the DE],
   35, 36.
3. W: course information handout; Quiz 1 Not! (try it later without the book 20-30 minutes max: then check answer key [hand, maple]);
   1.2: 1, 5, 15, 19 [Hint: write piecewise linear function from graph: v₁(t) for first expression, v₂(t) for second expression, solve first IVP for x₁(0) = 0, second IVP for x₂(5) = x₁(5)], 25 (like lunar landing problem), 39, 40 (ans: 2.4mi), 43.
4. F: 1.3:  3: [hand draw in all the curves on the printout supplied by bob (rotated and scaled to fit option on Print), maple version],
   8* : our first Maple HW problem: read the Maple HW instruction page, then try the Maple direction field command in this linked template to reproduce the two given curves by using two appropriate initial data points [Hint: notice that the y-axis crossing points are integers!] Then include all the initial data points of the red dots (some half-integer values); this is also described on p.29  for both Maple and Matlab, but the above template already has the appropriate section copied into it for you; if you can find a partner to work with in time fine, otherwise try this yourself and find a partner for future problems, then merge your work [will not be due until we finish chapter 1];
   being careful: 11, 15 [note D_yf is just df/dy above Eq (9) in the text]; 27 [don't worry about making a diagram];
   a freebie: numerical problems caused by vertical tangent line (figure 1.4.2).
   
   WEEK 2 [-1] no class M: Labor Day]
5. T: 1.4: 1, 5; 21, 25, 27, 29.
6. W: class list contact data sheet handout (not web published) to help form partnerships;
   Quiz 1;
   handout on exponential behavior/ characteristic time [read this worksheet explicit plot example];
   1.4: 45, 47 [recall 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;
   Optional challenge problems:
   68 (PHYS: parametric cycloid solution, obvious typo: θ = 2t in text, not θ = 2a) 
   or 69 (ME: suspension cable catenary: hyperbolic cosine)
   or Torricelli's law section and 64 (ChmE: ans r = 1/35 in).
7. F: quiz 1 answer key is online;
   no-paper handout, only on line: recipe for first order linear DE;
   1.5: 3, 7, 11, 21; 27;
   21*[check both the general solution and the initial value problem solution with the dsolve template].
   For future reference:
   > deq := y ' = x y      [space implies multiplication]
   > sol:=dsolve(deq, y(x))
   > solinit := dsolve({deq, y(0)=1}, y(x))
   [or enter DE plus IC separated by a comma, right click on output, choose Solve DE, for y(x), or Solve DE interactively]
   > y ' = x y , y(0)=1
   > odetest(sol,deq)   [this returns 0 if LHS-RHS of DE is 0]
   
   WEEK 3[-1]:
   Weeks 3 thru 4: come by and find me in my office, tell me how things are going.
   This is a required visit. Only takes 5 minutes or less. Come say hi!
    
8. M: ASK BOB ABOUT TEST 1 DATE;
   1.5: 17, 26 [ans: x = ½ y ^-2 +Cy ^-4],
   29 [use the fact that the derivative of the function defined by the integral is the integrand including all multiplicative factors], 31 [show by differentiation and backsubstitution into the DE, not by rederiving it],
   32 [find the constants by substituting the trial solution into the DE and choosing the values of A and B to make the DE satisfied: put all terms on LHS and set coefficients of the sine and cosine separately to zero],
   41 [Hint: Δt in years is the fraction of the year, so S(t) Δt is the approximate income during this time interval, 12 percent of which contributes to ΔA, while the interest contribution is the annual interest rate 6 percent prorated by multiplying by the fraction Δt for the interest contribution over this shorter time period, applied to the current value of the retirement amount A(t): 6 percent of A(t) Δt; adding these and dividing by Δt, taking limit to get dA/dt gives the linear DE [click here only if you have tried to get the DE and could not or its solution did not give the correct number].
   41* [use the dsolve template to solve this DE with its initial condition; evaluate answer to question of word problem to a floating point number and give the result in dollars and cents in a comment, then think about significant figures and give a response with fewer significant digits].
   When you are satisfied, compare with retirement.pdf.
   
   Maple1.mw assignment complete. Due week between Tuesday to next Monday.
   REMIND BOB to match up people who need Maple partners.
    
9. T: 1.5: 37 (Use Eq. 18 in the book; what is the final concentration of salt?);
   [this mixing problem is an example of developing and solving a differential equation that models a physical situation; for another word problem example see this]
   1.R(review): classify the odd problems 1-35 as: separable, linear in y (as unknown), linear in x (as unknown), some combination of these three, or NOTA (none of the above), for example dy/dx = y/x is all three and can be solved in three different ways;
   solve 25, express K in terms of C as given in the book supplied answers by combining the two terms in the K solution [use the identity for (x+1)³ !] and then comparing with the C solution;
   solve 35 in two ways and compare the results;
10. W: Quiz 2;
    Optional Light Reading 1.5: Application: see how a slight generalization of the directionfield example 1.3.3 to include an initial time parameter and time and temperature scale parameters has a useful physical application on pp.56-58. No need to study it, but reading through it gives an idea of how one can use a simple linear DE to model an interesting problem.
    Work on Maple assignment. Read Maple Tips to Remember on Examples and Tips page.
    Try the Maple Help menu Quick Tour: Ten Minute Tour + Numeric and Symbolic Computations.
11. F: handout on solution of logistic DEQ (and similar eqns) [directionfield,  integral formula];
    [handout on DE's that don't involve the ind var explicitly];
    2.1: 5 [go thru the solution steps just this once with concrete numbers, remember the integral formula, follow the steps of the book or handout derivation of the solution, compare with the final logistic formula; use the formula for the remaining HW problems],
    9, 15 [note that by definition B_o = kMP_o and  D_o = -kP_o² , 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 needed to solve problem 16],
    16 [ans: P = .95 M after 27.69 months].
    
    WEEK 4[-1]: Maple1 due this week, bob office visit this and next week; check Blackboard quiz grades
12. M: 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 β_o = 0.3 from condition dP/dt(0) = 3x10^5 using the DE at t =0, then use solution α=0.3915 from second condition (solve this yourself or at least check numerically that it satisfies the 6 month condition) to find the limiting population as t goes to infinity (i.e. neglect decaying exponential)].
13. T: 5:30pm voluntary Test 1 problem review session in MLRC;
    air resistance handout [comparison of linear, quadratic cases; numerical solution];
    2.3: 1, 2, 3, 9 [remember weight is mg, so mg = 32000 lb determines m = 1000 in USA units]. Homework is for Friday of course!
14. W: Test 1 on chapter 1: directionfields, initial data, first order linear DEs and separable DEs.
15. F: Open Maple at class beginning;
    2.4: most basic numerical DEQ solution technique from calculus: maple2.mw; in class
    open MAPLE, then open IE and webmail, use email link to go to our HW page in a separate window and follow instructions for this maple assignment;
    hand work home work:
    2.4: 1 (do by hand and calculator, can check with euler.mw if you wish;
    optional read about improved Euler).
    
    WEEK 5[-1]:
16. M: 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;
    Maple2.mws due next week September 29-October 2.
    
    > {x - y = 2, x + y = 0}  Right-click and choose solve
     
17. T: handout on RREF (Reduced Row Echelon Form, section 3.3);
    3.2: 1, 3, 7; 11, 13, 15 (do a few by hand, then you may use step-by-step  row ops with MAPLE or a calculator;
    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?]
     
18. W: Test 1 back: rules of algebra NOT!: 1/(x+y) = 1/x + 1/y ; check your online grades for accuracy;
    handout on  solving linear systems example;
    with a partner use the LinearSolveTutor and enter matrix 3.2.15 and reduce, then solve the system (if possible), then switch positions and enter matrix 3.2.18 and repeat; copy the commands below and try them on the second matrix;
    3.3: 1, 3, 7, 11, 17,  23 = 3.2.13;
    19* [use the tutor and record your step by step reduction, annotating each step as in the rowredex0.mw worksheet; then check with ReducedRowEchelonForm];
    To solve a linear system directly in Maple, input augmented matrix with the palette or Matrix commandand these 4 lines will do it:
    > with(Student[LinearAlgebra]):
       A:=Matrix([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);  [you enter matrix using palette instead]
       ReducedRowEchelonForm(A);
       BackwardSubstitute(%);
    To step by step reduce a matrix and solve the system do (pick Gauss-Jordan):
    > LinearSolveTutor(A)
19. F: Quiz 3 on HW discussed Wednesday;
    recall 3.2.23 above never discussed;
    3.2: 22  [solve using row ops on augmented matrix to rref form, then by hand backsubstitute, check answer with Maple];
    3.3: 29 = 3.2.19 [solve using row ops on augmented matrix to rref form, then by hand backsubstitute];
    29* [now check this same problem solution by solving with ReducedRowEchelonForm, BackwardSubstitute of augmented matrix after inputting the Student Linear Algebra package from the Tools menu];
    MAPLE chemical reaction problem*
    
    WEEK 6[-1]: Maple2 due this week.
20. M: word of the day: can you say "homogeneous"?;
    3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43
21. T: 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, matrix inverse, determinant, transpose
     > A B [note space between symbols to imply multiplication]
     > A^-1
     > |A|  [absolute value sign gives determinant]
     > A^%T
     
22. W: Quiz 4; read on-line handout: determinants and area etc;
    3.6: (determinants abbreviated: forget about minors, cofactors, only need row reduction evaluation to understand)
    13, 17, 21 (plug in just so you've done it once);
    17* [record your Gaussian elimination steps in reducing this to triangular form and check det value against your result];
    3.5. Apps:  7* from textbook on p.197, using the inverse matrix as described on p.196,
    be sure to answer the word problem question with a text comment;
    are these sandwiches cheap or not?
    
    maple3.mw assignment is complete; due next week (or at least the week after break if you have too much midterm stuff going on); get it over with early, catch up with others if not done yet [blowing off maple means subtracting 10 points from your cum for midterm grades: maple1 and maple2 only for midterm grades];
     
23. F: 4.1: 1; 5, 7; 9; 15, 17; 19, 23; 25 (use technology in 1) evaluating the determinant  or 2) ReducedRowEchelonForm needed for these problems);
    [HW row reductions];
    hand out on the interpretation of solving linear homogeneous systems of equations: A x = 0 [optional read: visualize the vectors]
    
    WEEK 7[-1]: maple3 due by week after break
24. M: Columbus Day?
    handout on nonstandard coordinates on R^2 and R^3 [goal: understand the jargon and how things work];
    1) Using the 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,with the blank graph paper handed out in class and 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 [-5,1] and find its new coordinates, then confirm by matrix multiplication. Similarly read off the old coordinates of the point whose new coordinates are [2,-1] and then confirm their relationship using matrix multiplication. Write down the coordinate transformations between the old and new coordinates in both directions. Put your name on it and bring it to class.
25. T: study the handout on solving linear systems revisited;
    [note only a set of vectors resulting from a linear homogeneous condition A x = 0 can be a subspace of a vector space; interpretation: lines, planes, ..., hyperplanes through the origin];
    4.2: 1, 3, 11; 15, 19;
    look at the soln to the system 3.4.21 on p.182 (back of book p.712): 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].
26. W: Quiz 5;
    handout on linear combinations, forwards and backwards [maple to visualize];
    4.3:  1, 3, 5, 7; 9, 13; 17, 21; 23.
27. F: handout summarizing linear vocabulary; [maple illustration of how to describe a plane in space]
    4.4: 1, 3, 5, 7; 9 [find solution, pull apart to find coefficient vectors of parameters, repeat for remaining 4 problems],
    13; 15, 17, 25 [use technology for all HW row reductions and determinants (2x2 dets are easy by hand!)].
    FALL BREAK  :-) Enjoy. Be safe in your travels.
    
    WEEK 8[-1]:  Test 2 W in week 9; maple3 due this week
    welcome back from fall break!
    [U calculate your own midterm Grade (check WebCT on-line grades)]: 4 quizzes (drop lowest), 1 test, 2 maple assignments; if you want better Maple grade, check how 4/4 =10 points of your cum makes a difference for you. Corrected worksheets must arrive by Monday after break for consideration for midterm grade.
     
28. M: catch up on 4.4 hw;
    Transition back to DEs:
    read p.274 on function spaces and examples 3, 6, 8, 9; equivalently read the worksheet on the vector space of quadratic functions [quadratics.mw];
    4.7: 15 [Hint: solve c₁(1+x)+c₂(1-x)+c₃(1-x^2) = 0, for unknowns c₁,c₂,c₃; are there nonzero solns? if not, these are lin. ind. 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 [same approach].
29. T: handout on sinusoidal example;
    M: 5.1: 1, 3, 5, 9; 13, 17;
    begin maple5.mw: 9* [use the 2nd order dsolve template to solve the 2nd order IVP as a check].
    > deq:= y'' + 3 y' + 2 y = 6 e^x
    > inits:= y(0) = 4, y'(0) = 5
    > solgen:=dsolve(deq,y(x))
    > solivp:=dsolve({deq,inits},y(x))
    or right click on output of (use solve DE interactive, don't forget set braces):
    > { y'' + 3 y' + 2 y = 6 e^x , y(0) = 4, y'(0) = 5}
30. W: 5.1: 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 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?].
31. F: Quiz 6;
    read the handout on visualizing the initial value problem (IVP);
    read handout on complex arithmetic, exponentials [maple commands; i is uppercase I  in Maple];
    5.2: 13, 17, 21, 26 [this problem now for fun now, later a method].
    
    WEEK 9[-1]:
32. M: [optional web only handout for context and going beyond this course: power series and DEs (chapter 11)];
    handouts on the amplitude and phase shift of sinusoidal functions and exponentially modulated sinusoidal functions [maple videos showing different phase shifts and frequencies]; do the short problems at the end of each handout after studying them;
    5.3: complex roots:
    8 [ans: y =  exp(3x)(c1 cos(2x) + c2 sin(2x))] , 9; 17; 22, 23 (express in phase-shifted cosine form);
    use technology to factor or solve for roots of polynomial equations:
    > solve(r²+6 r+13 = 0)    (in general, finds all roots exactly up to fourth degree and sometimes higher degree if lucky)
    > factor(r²+6 r+13 = 0)    (in general, factors real polynomials into linear and irreducible quadratic factors)
    [Note: > fsolve(...) returns all numerical roots of a polynomial].
33. T: voluntary MLRC problem session 5:30pm;
    HW for Friday:
    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].
34. W: Test 2 on chapters 3, 4.
35. F: 5.4: 1, 3, 13, 14 [b) the solution once put in the form x = A exp(-Kt) cos(ω t - δ) has the "envelope curves" x = +/- A exp(-Kt) ] , 17; 23 [pdf].
    handout on linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillator) [on-line only: examples in nature].
    
    WEEK 10[-1]:
36. M: 5.4: 21 (solve by  hand), 21* [solve the two DEs with the same initial conditions using Maple and plot the two solution curves together, but also plot the corresponding third solution with pure damping but no oscillator as in the template];
    handout on driven (nonhomogeneous) constant coeff linear DEs
    [complete final exercise on sheet [solution], Maple: *check with dsolve];
    5.5: 1, 3, 9 [we will not cover "variation of parameters"; the book presentation of the method of undetermined coefficients is a recipe with little justification (educated guess), instead the handout shows exactly how and why one gets the particular solutions up to these coefficients].
37. T: handout on damped harmonic oscillator driven by sinusoidal driving function [Maple specific example]
    [maple resonance plots: general , specific (frequency much higher than resonance)];
    5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85 - (6 cos(3x)+7 sin(3x))/85]; [not many of the book driving functions are physically interesting here]
    5.6: 8. Ignore the book instructions and solve this problem by hand for the initial conditions x(0) = 0 = x'(0). Read the worksheet about the interpretation of the solution.
38. W: Quiz 7 thru Monday HW [> x, -x, x sin(x)    Enter and right click on output to plot together!]
    handout on beating and resonance
    [Tacoma Narrows Bridge collapse: engineering explanation; Wikipedia bridge video; Google];
    maple5.mw due next week (5 problems now complete);
    5.6: 1 [rewrite as a product of sines using the cosine difference formula:
    cos(A) - cos(B) = {-2 sin((A-B)/2)} sin((A+B)/2), the expression in {} is the sinusoidal amplitude, look at the plot of x and +/- this amplitude function together, then repeat for the HW problem* at the end of the worksheet and include only that HW problem section in your maple5 worksheet],
    11 [convert steady state solution to phase shifted cosine at end of calculation], 17 [pdf], 23.
39. F: recall coupled system of DEQs [now revisit it (paper handout giving a preview of what we are about to embark on)];
    Watch the MIT Eigenvector 3 minute video [there are 5 frames which then repeat, so stop when you see it beginning again---from their LinAlg course]; see the 5 possibilities of the video in this Maple DEPlot phaseplot template;
    if you like, 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 (try on HW matrices!), click on end of red vector and drag around an approximate circle to see corresponding blue vector];
    6.1: 1, 9 only 2 problems so you have time to do some of the above!
    
    WEEK 11[-1]:
40. M: 6.1: 13, 19; 27, 25;
    [go thru process: use determinant: |A-λI|, solve (solve if complex roots) to find characteristic equation and eigenvalues of a 3x3 or higher dim matrix];
    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 = BY and Y = B^-1X to find the new coordinates of the point [x1,x2] = [-2,4] in the plane and to find the point whose new coordinates are [y1,y2] = [2,-1]; finally evaluate the matrix product by hand A_B = B^-1AB to see that it is diagonal and has the corresponding eigenvalues in order along the diagonal. [Remember this? (the geometry of diagonalization)]
     
41. T: 6.2: 1, 9 ("defective"); 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"?];
    handout on the geometry of diagonalization and 1st order linear homogeneous DE systems (2d example: real eigenvalues [phaseplot]);
    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>;
     
42. W: Quiz 8 on resonance (5.6 resonance calculation, like a particular solution part of 5.6.17 ignoring the homogeneous part and initial conditions);
    solve the 3x3 system x' = A x for the matrix of problem 6.1.19, with initial condition x(0) = <2,1,2>;
    handout on 1st order linear homogeneous DE systems (2d example: complex eigenvalues [phaseplot]);
    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;
    handout on transition from a complex to a real basis of a linear DE solution space.
     
43. F: maple5 is due (please submit soon);
    handout on 1st order linear homogeneous DE systems (2d example:  purely imaginary eigenvalues);
    using the eigenvector technique, find the general solution for the DE system x ' = A x for the matrix A = <<1|-5>,<1|-1>> (input by rows) and then the solution satisfying the initial conditions x(0) = <1,2>; note the solutions are easily obtained with dsolve (*check your hand solution this way!);
    Read 7.3 [especially compartmental analysis example 4, which has complex eigenvectors (more examples)];
    do 7.3: 37 [closed 3 tank system with oscillations (explanation), plug into Eq(22), solve by eigenvector method].
    
    WEEK 12[-3]:
44. M: 5:30 MLRC Takehome Test 3 problem session; have you submitted maple3 and maple5?
    Read 7.1 except for examples 5-7 (opposite of reduction of order, not needed);
    read handout on reduction of order (optional on-line handout on phase spaces);
    7.1: 1, 8 [first let   = [x1,x2,x3,x4] = [x,y,x',y'], then re-express the 2 DEs replacing x1" by x3' and x2" by x4', adding the definitions x1' = x3, x2' = x4, then write the 4 DEs in matrix form x ' = A x + F],
    7.2: 5, 9.
    These are very short problems, no solving required, just rewrites, to make sure you understand matrix notation.
     
45. T: on-line summary of eigenvector solution of a linear homogeneous system of differential equations
    
    Thanskgiving Break. Be safe in your travels.
    
    WEEK 13[-3]: Test 3 on chapters 5, 6, and eigenvector solution of first order homogeneous DE systems [don't procrastinate, you will regret it]
46. M: Work on Test 3 [no class];
    Take home test 3 out, due back in one week Monday night; Honor system to be respected.
    [on-line PDF linked here Sunday; 2-sided paper copy on my office door Sunday for pickup Monday morning or later];
    Read test instructions.
    Make a BIG push on it over next 4 days when you will have no math HW.
    Think of this take-home test as an exercise in "writing intensive" technical expression. Try to impress me as though it were material for a job interview (you're fired! or you're hired! ?). In a real world technical job, you need to be able to write coherent technical reports that other people can follow.
47. T: Work on Test 3 [no class]
48. W: Work on Test 3 [no class]
49. F: bob returns; if you have questions, today is the day to find bob in his office;
    homework returns. Make sure you have read about reduction of order and done associated short homework on it.
    Redo 5.2 Example 6 ( .mw ) using reduction of order (x₁, x₂) = (y, y ') and the eigenvector solution of the 2x2 DE system, just like our worked example in class
    [here is another similar example with details of solution for undetermined coefficients (.mw)]
    
    WEEK 14[-3]:
50. M: Test 3 due by tonight anytime, or Tuesday in class, stapled please.
    note the higher order system dsolve template
    > {x1''(t)=x2(t), x2''(t)=x1(t), x1(0)=1, x1'(0)=0, x2(0)=0, x2'(0)=0}   [enter, right click and use Solve DE Interactive]
    handout on extending eigenvalue decoupling (to nonhomogeneous case, and second order);
    handout on reduction of order worked example: solution;
    solve x''(t) = A x(t) + F(t), where x(t) = <x1,x2>, F(t) = <0,cos(t)>, A = <<0,4>,<4,0>>, x(0)=<1,0>, x'(0)=<0,0> [solution: .mw, eigenvector details].
     
51. T: handout on 2 spring 2 mass system; solve the stated initial conditions for the system initially at rest at equilibrium to obtain the result on the handout [fun maple design plot template];
    find the new particular solution for F(t) = <0,cos(2^1/2 t)>; since the three periods have irrational ratios, this motion is not periodic, if you use the template to visualize the plots of solutions for any initial conditions, you will see interesting behavior as you increase the time up to 100 π .
    [You can read about this from the textbook perspective in section 7.4 up to the resonance subsection.]
     
52. W: solve x''(t) = A x(t) + F(t), where x(t) = <x1,x2>, F(t) = <0,cos(2t)>, A = <<-5,2>,<2,-2>>, x(0)=<0,1>, x'(0)=<0,0> (pdf, .mw);
    use the method of reduction of order to solve x''(t) = A x(t) + F(t), where x(t) = <x1,x2>, A = <<0,4>,<4,0>>, x(0)=<1,0>, x'(0)=<0,0> (same system as day 50 without driving force: soln pdf)
     
53. F: 7.4: (3 spring 2 mass system) 3, 9 (undriven/driven 3 spring system)
    [7.4.9 continues 7.4.3; can use direct substitution of x1 = c5 cos(3t), x2 = c6 cos(3t) into DEs to determine undetermined coefficients c5,c6 since there is no damping term to mix cosines and sines; also solve the IVP with inits: x(0) = 0, x'(0) = 0: .mw];
    
    reduction of order will not be used to treat damping on the final exam, but if you are interested in the finishing touches to this approach of handling coupled damped harmonic oscillator systems, read the final section of the worksheet on the 2 mass 2 spring system or of the 2 mass 3 spring system [note that these systems without damping can be solved for arbitrary values of the system parameters just using the quadratic equation and 2x2 matrix linear system solution]
    
    Final WEEK:
54. M: All  Test 3 late submitters due today or never;
    consider Final Exam 07S.
55. T [F schedule]: Test 3 back with answer key. Discussion.
56. W: CATS evaluation.

I try to keep a class or so ahead with the homework for those of you who might anticipate?

Weeks 3 thru 4: come by and find me in my office, tell me how things are going. This is a required visit. Only takes 5 minutes or less.

*MAPLE homework log and instructions [asterisk "*" marked homework problems]

Test 1: ; MLRC problem session .
Test 2:  ; MLRC problem session .
Test 3: Take home out , in  ; MLRC problem session .
Final Exam: MLRC problem session

FINAL EXAM: if convenient, you may get permission to change section (email bob)
     2705-01:   1:30 MWF/1:30T:  Tue, Dec 18 1:30 - 4:00
     2705-06: 10:30 MWF/11:30T: Sat, Dec 15 1:30 - 4:00

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

extras

• MIT linear algebra on-line videos
• Duke University on-line linear system interactive applets
• earthquake building vibration modeling
• the 2 spring system with damping and 2 resonance peaks: 2spring.mw
• Find general solution: x' =  A x + f  with A = <<2,4>,<5,1>>, f = <3 exp(t), -t²>, using method of undetermined coefficients on decoupled equations.