In class: log on to Novell [G90] and open Internet Explorer. (IE allows you to open MAPLE files linked to web pages automatically if MAPLE is already open, in Netscape you must save the file locally and then open in it MAPLE.) Open VU Application Explorer through its desktop icon and choose Academic applications, Math&Stat to find MAPLE. bob will quickly show you the computer environment. [We will edit your internal VU homepage righthand side links (bottom of column) to add a link to http://citrixweb.villanova.edu for access to Maple anywhere on the internet.]
In class: log on to the Villanova home page (click on the upper right "login" icon and use your standard VU email username and password) and check out our My Classrooms classroom site, and visit the link to my course homepage from it
[ http://www.homepage.villanova.edu/robert.jantzen/courses/mat2705/ ],
After class: 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) to return in class Thursday.
[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.]
Read computer classroom etiquette.
Homework Problems: 1.1: 3, 5, 13, 33 (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.
handout algebra/calc background sheet;
1:1: 7a [only check y1(x)], 23, 27, 29 [Hint: recall perp lines have slopes which are negative reciprocals, make a diagram of given point (0,1), curve, and point (x,y) on curve and connecting line segment between points perpendicular to tangent line, compute slope from two points, and from derivative, equate], 35, 36.
[Maple is a required part of the Calc1,2,3 syllabus. If you were shortchanged, or are a freshman who did not take one of these courses last semester, or both, there is no need to panic. Do sections (1-4)(6)(12) of the Maple Help Menu New User Tour to get acquainted, and/or stop by bob's office for a demo. All Maple assignments will be supported by templates and unlimited individual consulting with bob when necessary.]
1.2: 1, 5, 15, 19 [just do roughly, no need to use graph paper; life is too short], 25 (like lunar landing problem), 39, 40 (ans: 2.4mi), 43.
WEEK 2:
paper worksheet handed out for problem 1.3.3 (use a pencil and eraser to draw in the 12 curves);
1.3: 3; 8*: try the MAPLE direction field command in the section on first order DEs of cmdlist4.mws (open in IE after opening MAPLE or save locally and open in MAPLE, reproduce the two given curves by using two appropriate initial data points [Hint: y-axis crossing points are integers!]; this is also described in the technology supplement Computing Projects Manual for both MAPLE and MATLAB and after this section in your textbook, 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]; 11, 15 [note Dyf is just df/dy].
1.4: 1, 5; 21, 25, 27, 29.
handout on exponential behavior characteristic time;
1.4: 29, 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.
[Note WYSISYG Maple]
[wildcard photo release can be done by logging into VU homepage on web, then Edit Account, go to WildCard Permission link]
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 := diff(y(x),x) = x*y(x);
> sol:=dsolve(deq,y(x));
> sol_init := dsolve({deq,y(0)=1},y(x));
WEEK 3:
find maple partners please, first maple assignment will be due next week;
[Weeks 3 and 4: come by and find me in my office, tell me how things are going.
This is a required visit. It will take only 5 minutes or less.]
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];
1.R (review): classify the odd problems 1-35 as: separable, linear in y, linear in x, 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 and then comparing with the C solution;
solve 35 in two ways and compare the results.
read pp. 73-74 of 1.6. Computer Algebra Solutions to be aware of the very different form solutions can take from different approaches to solving a DEQ, but only see if Maple still gives the result quoted in the book when you type in the single dsolve command and execute it;
1.4.23 is both separable and linear (in both y and x!), solve it in both ways (separable, linear in y) and compare, then check your solution by backsubstitution into the DEQ;
catch up on previous hw to be ready for the quiz.
;handout on solution of logistic DEQ;
2.1: 5 [go thru the solution steps just this once with concrete numbers, remember the integral formula, follow the steps of the book or handout derivation of the solution, compare with the final logistic formula; use the formula for the remaining HW problems],
9, 15 [note that by definition Bo = kMPo and Do = -kPo2 , and these are easily solved for M and kM in terms of Bo, Do, and Po to get the other two logistic curve parameters needed to solve problem 16],
16 [ans: P = .95 M after 27.69 months];
WEEK 4: don't forget 5 minute office visit;
maple1.mws due Feb 7-13 by email, follow the instructions in the link, especially the ones about partners and subject heading and filename.
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],
30, 31 [ans: find βo = 0.3 from condition dP/dt(0) = 3x10^5 using the DE, then use solution α=0.3915 from second condition (check numerically that it satisfies the 6 month condition) to find the limiting population as t goes to infinity (i.e. neglect decaying exponential)];
handout on DE's that don't involve the ind var explicitly;
education is also culture enhancement: http://www.artsci.villanova.edu/eastasia/ChinaFilmweek.htm
2.3: 1, 2, 3, 9 [remember weight is mg, so mg = 32000 lb determines m = 1000 in USA units].
2.4: most basic numerical DEQ solution technique from calculus: maple2.mws; in class
open MAPLE, then open IE and webmail, go to our HW page in a separate window and follow instructions for this maple assignment.
hand work home work:
2.4: 1 (only do the Euler method not improved Euler method, do by hand and calculator, can check with euler.mws if you wish; optional try improved Euler).
handout: why LinAlg with DE?; [maple dsolve and DEplot for 2x2 systems of DEQs];
for Wednesday: read 3.1 on linear systems,
do 3.1: 1, 5, 7; 9, 15, 31;
Maple1.mws due.|
WEEK 5:
Test 1 through 2.1 [separable DEQ, linear DEQ, Newton's law of cooling word problem];
handout on rref and solving linear systems example;
3.3: 1, 3, 7, 11, 17, 23 = 3.2.13;
19* [check with rref];
to solve a linear system, input augmented matrix and do:
> with(linalg):
> A:=matrix([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);
> rref(A);
backsub(%);
handout on solving linear systems; [if interested, read about Maple options for doing this]
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 problem solution by solving with rref, backsub of augmented matrix];
meet with partner(s) to complete maple2.mws, start maple3.mws.
WEEK 6: maple2 due by wednesday [10 people have still not done the office visit thing...]
3.4: 1, 5, 7, 11, 13; 17, 21; 27, 43.
30 [just substitute for the inverse and then drop parentheses and multiply out: (ABC)-1 (ABC) = ...],
32 [multiply on the left by A-1].
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];
MAPLE problem 7* from textbook p.197 using the inverse matrix as described on p.196 [also Computing Projects Manual: 3.5.7 on page 74],
be sure to answer the word problem question with a text comment;
are these sandwiches cheap or not?
[if you know how to use MathCad, try it and compare with MAPLE]
maple3.mws is due next week, why not get it out of the way this weekend?
catch up on chapter 3.
WEEK 7: maple3 due this week [7 people have still not done the office visit thing...]
handout on nonstandard coordinates on R^2 and R^3 [goal: understand the jargon and how things work]
look at the soln to 3.4.21 (back of book): what does it tell us about the 4 vectors (columns of the coefficient matrix) in R2? how many independent relationships are there among the 4 vectors? How many vectors are independent? Is this obvious if you look at the explicit components of these vectors?
4.3: 1, 3, 5, 7; 9, 13; 17, 21; 23.
maple upgrades by Tuesday after break to affect midterm grade.
Spring Break.
enjoy and be safe.WEEK 8: Test 2 Friday April 1
handout summarizing linear vocabulary to recall what we were doing before break;
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!)];
from illustrated coordinate handout with new basis {[2,1],[1,3]} of plane: find new coordinates of point [3,14]; since this vector is off the diagram grid, if you take the half vector [3/2,14/2], and mark it off on the diagram and use a ruler to draw lines parallel to the new coordinate axes, do you get half those coordinates?;
read p.274 function spaces and examples 3, 6, 8;
4.7: 15 [solve c1(1+x)+c2(1-x)+c3(1-x^2) = 0, for unknowns c1,c2,c3; nonzero solns?]
5.1: 1, 3, 5, 9; 13, 17;
begin maple5.mws: 9* [use the 2nd order dsolve template to solve the 2nd order IVP as a check].
> deq:=diff(diff(y(x),x),x)+3*diff(y(x),x)+2*y(x)=6*exp(x);
[ or: deq:=(D@@2)(y)(x) +3*D(y)(x)+2*y(x)=6*exp(x); ]
> inits:=y(0)=4,D(y)(0)=5;
> gen_sol:=dsolve(deq,y(x));
> ivp_sol:=dsolve({inits,deq},y(x));
49 [find the IVP solution for y, set y ' = 0 and solve exactly for x, backsubstitute into y]
5.2: 1, 11;
5.1: 49*[check your IVP solution for the highest curve in Figure 5.1.6 using the dsolve template and plot it together with the horizontal line y=16/7 claimed by the book 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 as in this case; is 16/7 the peak value of your solution?].
5.2: 13, 17, 21, 26.
WEEK 9
5.3: 3, 8 [ans: y = exp(3x)(c1 cos(2x) + c2 sin(2x))] , 9; 17;
use technology to factor or solve for roots of polynomial equations:
> solve(r^2+6r+13=0);
> factor(r^2+6r+13);
21, 23;
[Note: > fsolve(...) returns all numerical roots of a polynomial].
49* [check your solution with the higher order dsolve template, and also check the IC linear system solution as in the template, good practice for being able to check solutions on test 2].
EASTER BREAK
Quiz 6 back;
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.
handouts on linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillator) and on the amplitude and phase shift of sinusoidal functions.
on the weekend, catch up with current Maple assignments.
1 homogeneous linear system of eqs, 1 non-homogeneous system of eqs,
1 higher order homogeneous linear constant coefficient deq with initial conditions (IVP).
WEEK 10
handout on RLC circuits [plots for this example];
5.4: 15, 21;
underdamped RLC circuit parameters: R = 16 ohms, C = 1/40 = .025 = 25 millifarads, L = 8 henries, E0= 17 volts; evaluate τ0, ω0, T0, ω0 τ0, τ, ω, T, ω τ, A0 = E0/(ω L), and I(t) from the handout formulas.
Maple: *[Make a plot showing the current and its amplitude envelope for one decay time t = 0..τ and 4 decay times for this new problem, as in the example plots, but only two figures.]
5.5: handout on driven constant coeff linear DEs
(complete final exercise on sheet, Maple: *check with dsolve);
5.5: 1, 3 (we will not cover "variation of parameters"; the book presentation of the method of undetermined coefficients is a recipe with little justification, instead the handout shows exactly how and why one gets the particular solutions up to these coefficients).
[maple plots: general specific];
repeat the undriven RLC problem of day 37 now with a voltage source E(t) = 4 sin(2 t) and initial conditions I(0) = 0 = I '(0), solving it ignoring the solution formulas derived on the handout; then check your result with these formulas.
5.5: 33, 38 [y = ( exp(-x)(176 cos(x)+197 sin(x)) )/85 - (6 cos(3x)+7 sin(3x))/85].
*maple plot: for yesterday's RLC problem, plot I, Ip (the steady state part of the solution) and E/L on the same plot and see how long it takes for the transient to essentially be zero to the pixel accuracy;
handout on beating and
resonance [engineering explanation] [bridge clip, only seems to work in Netscape];
maple5.mws due week April 11-15.
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, try plotting x and +/- this amplitude function together],
11 [convert steady state solution to phase shifted cosine at end of calculation], 17, 23.
WEEK 11: maple5.mws due this week
recall coupled system of DEQs;
6.1: 1, 9, 13, 19; 27, 25;
[use det, factor, solve (solve if complex roots) to find characteristic equation and eigenvalues of a 3x3 or higher dim matrix].
2) Find an (integer) eigenvector basis changing matrix B = augment(b1,b2,b3) for the matrix of 6.1.19 which is the result of your hand method yesterday and use it to find the new coordinates [y1,y2,y3] of the point with old coordinates [x1,x2,x3] = [2,1,2]; how does the basis you wrote down compare with the eigenvectors that MAPLE finds or which is given in the back of the book?; evaluate AB = B-1AB.
13* [check with eigenvects command, choose eigenvector basis matrix B, evaluate AB = B-1AB by matrix multiplication].
Test 2 back with answer key;
handout on solving first order linear homogeneous constant coefficient systems of differential equations;
find the general solution for the DE system x ' = A x for the matrix A = matrix([[1,4],[2,3]]) from Th HW, then the solution satisfying the initial conditions x(0) = [-2,4]; repeat for the matrix of 6.2.1 and the initial condition x(0) = [0,1]; *check your solution for the first matrix using the dsolve system template; repeat for the matrix of problem 6.1.19, with initial condition x(0) = [2,1,2].
WEEK 12:
1) Find the general solution for the DE system x ' = A x for the matrix A = matrix([[0,4],[-4,0]]), then the solution satisfying the initial conditions x(0) = [1,0];
repeat process for problem 7.3.9 (solution on-line).
MLRC problem session W, 5:30pm
on chapter 5 plus eigenvector method for solving 1st order linear homogeneous constant coefficient linear systems of deqs (6.1-2, 7.3);
Think of the 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. In a real world technical job, you need to be able to write coherent reports that other people can follow;
in class: Peer-to-Peer tutoring session based on HW problems as test prep [click on this link].
If you miss class, the test will be on-line at the usual place by about 3:30pm.
7.1: 1, 8 [first let [x1,x2,x3,x4] = [x,y,x',y'], then re-express the 2 eqns replacing x1" by x3' and x2" by x4', adding the definitions x1' = x3, x2' = x4, then write the 4 eqns in matrix form x ' = A x + F],
7.2: 5, 9.
These are very short problems, no solving required, just rewrites.
WEEK 13:
to finish Wednesday with your original group.
read section 7.4 of textbook.
really read section 7.4, no kidding.
WEEK 14:
7.3.35 [closed 3 tank system, no oscillations]; 7.3.36 [closed 3 tank system, oscillations].
7.4: 3, 9 [also solve the IVP with inits: x(0)=0, x'(0)=0 ; continues 7.4.3; can use direct substitution of x1 = c5 cos(3t), x2 = c6 cos(3t) into DEs to determine c5,c6 since there is no damping to mix cosines and sines];
then explore resonance by replacing the frequency 3 by ω and recomputing the particular solution and evaluating its amplitude function (sqrt of the sum of squares of the individual amplitudes); plot it and find the values of ω where A(ω) goes infinite (since there is no damping); these are the resonant frequencies;
recall from 7.4.3 (7.4.1 backreference?) the equations are:
x1 " = -(k1+k2)/m1*x1+ k2/m1*x2
x2 " = k2/m2*x1 -(k2+k3)/m2*x2 + F(t)/m2
(try problem 2 with x" = - A x, add the initial extra condition x'(0) = [0,0,0]);
[Test 3 answer key back today, but grading will take a week!]
Review problem session in class; come with any problems you want worked;
remember how to use Maple as a check in solving systems of equations or differential equations;
CATS;
5:30 MLRC review problem session (optional as usual).
F: bob in office 7:30-9:55, 11:30-3 approximately
M: bob in office 7:30-1:00 approximately
Final exams Sa, M [see below] [Sat exam can be delayed to M with bob's permission]