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 10 Standard (red not yellow icon) from the Start Menu Program listing under Math Applications
[or click on this maple file link: maple1.mw]
3) bob will quickly show you the computer environment supporting our class.
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 Wednesday.
[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.]
Homework Problems: 1.1: 3, 5, 13, 33 [ "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 Maple10 Standard DE entry and "odetest" for problems 3,5,13 [even 33!]
[use citrixweb Maple on your own computers]
handout algebra/calc background sheet;
1:1: 7a [only check y1(x)], 23;
formulating DEs: 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.
1.2: 1, 5, 15, 19 [just do roughly, no need to use graph paper; life is too short; actually one can do it analytically, see back of book], 25 (like lunar landing problem), 39, 40 (ans: 2.4mi), 43.
WEEK 2 (-1):
1.3: 3; [hand draw in all the curves on the printout supplied by bob (rotated and scaled to fit option on Print)],
8*: our first Maple HW problem: read the Maple HW instruction page (above), 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.
being careful: 11, 15 [note Dyf is just df/dy above Eq (9) in the text]; 27 [don't worry about making a diagram].
1.4: 1, 5; 21, 25, 27, 29.
handout on exponential behavior/ characteristic time [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: theta = 2t in text)
or 69 (ME: suspension cable catenary: hyperbolic cosine)
or Torricelli's law section and 64 (ChmE: ans r = 1/35 in).
[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 := y ' = x y [space implies multiplication]
> sol:=dsolve(deq, y(x))
> solinit := dsolve({deq, y(0)=1}, y(x))
WEEK 3 (-1): maple1.mw (3 short problems) due week of W Feb 1 to T Feb 7
[read the Why Rules Are Important for Submission page]
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].
1.5: 37 Mixing problem (Use Eq. 18 in the book; what is the final concentration of salt?);
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 [identity for (x+1)^3 !] and then comparing with the C solution;
solve 35 in two ways and compare the results.;
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.
Mandatory 5 minute Office (come say hi)
visit for all students during next two weeks (preferably before Test 1!);
Quiz 2 answer key on-line with Maple Worksheet;
handout on solution of logistic DEQ [directionfield, integral formula];
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];
maple1.mw due by Tuesday night;
handout on DE's that don't involve the ind var explicitly;
2.1: 6 [ use handout integration formula with sign reversed, or use technology for integral and combine log terms;
answer: > dsolve({diff(x(t),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],
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 (check numerically that it satisfies the 6 month condition) to find the limiting population as t goes to infinity (i.e. neglect decaying exponential)].
WEEK 4 (-1):
2.3: 1, 2, 3, 9 [remember weight is mg, so mg = 32000 lb determines m = 1000 in USA units].
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).
Read 3.1 on linear systems (review of HS 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 February 13-19.
Th: 5:30 MLRC voluntary problem review session for Test 1.
checking a soln of a DE or imposing conditions on a family of functions to make them solns of a DE, solving DEs which are separable or linear in either the independent or dependent variable (and recognizing N.O.T.A.), initial value problems, handling additional conditions to determine unknown parameters in the DE, understanding a directionfield.
WEEK 5 (-1):
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?];
in class with neighbor as partner, read together and execute line by line this linked worksheet: rowredex0.mw;
handout on RREF and solving linear systems example;
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, input augmented matrix with the palette or Matrix command:
> with(Student[LinearAlgebra]):
A:=Matrix([[3,8,7,20],[1,2,1,4],[2,7,9,23]]);
ReducedRowEchelonForm(A);
BackwardSubstitute(%);
To step by step reduce a matrix and solve the system do (pick Gauss-Jordan):
> LinearSolveTutor(A)
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];
MAPLE chemical reaction problem*;
consult the matrices section of the cmdlist4.mw worksheet for how to work with matrices in Maple;
Don't forget: Maple2.mw is due this week.
Not all Maple1.mw files have been received! These are easy points.
Your grade starts down from 90 instead of 100 if you blow them off.
If you got less than 80 on Test 1 (back today), you should come for an office chat.
WEEK 6(-1):
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
> A^(-1);
> A-1
> with(Student[LinearAlgebra]):
> Determinant(A)
> A%T
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];
3.6. 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?
[if you know how to use MathCad, try it and compare with MAPLE]
Maple3.mw assignment complete; due next week; 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...]
[row reductions];
handout on nonstandard coordinates on R^2 and R^3 [goal: understand the jargon and how things work]
and on the interpretation of solving linear homogeneous systems of equations: A x = 0 [.mw]
WEEK 7(-1):
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]
4.3: 1, 3, 5, 7; 9, 13; 17, 21; 23.
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 the old illustrated coordinate handout with 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 do the reverse for the point whose new coordinates are [2,2].
Transition back to DEs:
read p.274 on function spaces and examples 3, 6, 8, 9; equivalently a handout on the vector space of quadratic functions [.mw, 1.5MEG PDF];
4.7: 15[solve c1(1+x)+c2(1-x)+c3(1-x^2) = 0, for unknowns c1,c2,c3; 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];
Spring Break.
enjoy and be safe.WEEK 8(-1): Wed noon midterm grades, Maple upgrades for midterm grade possible until early Wed am
begin maple5.mws: 9* [use the 2nd order dsolve template to solve the 2nd order IVP as a check].
> deq:= y'' + 3 y' + 2 y = 6 ex
> inits:= y(0) = 4, y'(0) = 5
> solgen:=dsolve(deq,y(x))
> solivp:=dsolve({deq,inits},y(x))
handout on yesterday's sinusoidal example;
5.1: 33, 35, 39,
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; is 16/7 the peak value of your solution?].
feedback says Monday, March 27, with MLRC review on Th: 23;
handout on visualizing the initial value problem (IVP);
handout on complex arithmetic, exponentials [maple commands; i is uppercase I in Maple];
5.2: 13, 17, 21, 26.
handouts on the amplitude and phase shift of sinusoidal functions and exponentially modulated sinusoidal functions [maple videos];
Quiz 6;
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].
WEEK 9 (-1):
5.3[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 PLEASE, good practice for being able to check solutions on test 2].
handout on linear homogeneous 2nd order DEQ with constant positive coefficients (damped harmonic oscillator) [examples in nature].
metric system prefixes;
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 (either resolve the IVP with Maple or plug into the formulas for the solution);
Maple: *[RLC Maple plot: Make a single plot showing the current and its amplitude envelope for 4 decay times t = 0.. 4τ for this new problem.]
Th 5:30 MLRC Test 2 problem review session
5.5: handout on driven constant coeff linear DEs
(complete final exercise on sheet, 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, instead the handout shows exactly how and why one gets the particular solutions up to these coefficients).
WEEK 10(-1):
[maple resonance 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];
*RLC Maple plot: for this RLC problem, plot I, Ipart (the steady state part of the solution) and E/(2L) on the same plot and see how long it takes for the transient (difference between these two currents) to essentially be zero to the pixel accuracy.
[USA rms voltage 120 Volts means peak voltage 120 sqrt(2) = 169.71V]
maple5.mw due next week;
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.
if you like play with the Duke U applet [red is x, blue is A x, click on matrix entries to change];
see the various possibilities of the video in this Maple DEPlot phaseplot template;
recall coupled system of DEQs [now revisit it (preview of what we are about to embark on)];
6.1: 1, 9, 13 only 3 problems so you have time to do some of the above!
[use det, factor, solve (solve if complex roots) to find characteristic equation and eigenvalues of a 3x3 or higher dim matrix].
Sa:
April Fools Day!
►WEEK 11(-1):
For the matrix A = Matrix([[1,4],[2,3]]) entered by rows, A = <<1,2>|<4,3>> entered by columns,
find a set of independent integer component eigenvectors {b1,b2} by hand and then check them with the MAPLE Student[LinearAlgebra] Eigenvector command, then make the basis changing matrix B = <b1|b2> whose columns are these vectors (so we can agree, choose their order so that one gets to the second from the first by moving counterclockwise in the plane by an angle less than 180°), and write out the two coordinate transformations X = BY and Y = B-1X explicitly in terms of the individual coordinates: old [x1,x2] and new [y1,y2]; use the either the matrix or scalar transformation equations to find the new coordinates of the point [-2,4] in the plane and to find the point whose new coordinates are [2,-1]; then make a hand (print out some graph paper) or MAPLE plot printout of the new coordinate unit grid for the range y1 = -2 .. 2, y2 = -2 .. 2 and mark these two vectors on the plot and confirm visually the relationship with the coordinates you evaluated; finally evaluate AB = B-1AB. [bring this to class to give to bob]
6.2: 1, 9 ("defective"); 13, 17, 21 ("defective"); 34 [just write out the characteristic equation and solve with the quadratic formula, think about real distinct roots];
13* [check with Student[LinearAlgebra] Eigenvectors command, choose eigenvector basis matrix B, evaluate AB = B-1AB by matrix multiplication].
repeat this exercise for the matrix of 6.2.1 (namely A = Matrix([[5,-4],[2,-1]]) input by rows) and the initial condition x(0) = [0,1];
*check your solution using the dsolve system template;
repeat by hand for the matrix of problem 6.1.19, with initial condition x(0) = [2,1,2].
handout on 1st order linear homogeneous DE systems (complex eigenvalues) [phaseplot];
Find the general solution for the DE system x ' = A x for the matrix A = Matrix([[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 7.3.9 (solution on-line [.mw, .pdf]).
WEEK 12(-1):
finally express the component functions for this IVP solution in phase-shifted cosine form by hand.
handouts on reduction of order and extending eigenvalue decoupling (to nonhomogeneous case, and second order);
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;
optional handout on phase spaces, and changing from a complex to a real basis of a solution space.
Easter Recess: F, M no class
Week 13(-3):
Read 7.3; 7.3: 37 [closed 3 tank system with oscillations, plug into Eq(22), solve by eigenvector method].
on chapter 5 plus eigenvector method for solving 1st order linear homogeneous constant coefficient linear systems;
read test rules;
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 (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.
7.4: 3, 9 (undriven/driven 3 spring system)
[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 term to mix cosines and sines].
Week 14(-3):
recall that the equations in standard form are:
x1 " = -(k1+k2)/m1 x1+ k2/m1 x2 + F1(t)/m1
x2 " = k2/m2 x1 -(k2+k3)/m2 x2 + F2(t)/m2
handout on second order system with resonance exploration.
7.4.15;
Reformulate 7.4.9 as a first order system of DEs for the 4 component initial condition (state) vector X = <x ,x'> and solve it by the eigenvector technique, comparing with the previous direct solution as a second order system
No homework reward for hard work on Test 3; catch up on old Maple if you are behind.
Week 15( -3):
for homework, review Final Exam 05S from the archives.
teaching evaluation forms [CATS!
];almost goodbye.
Final Exam Thursday May 11 4:15-6:45pm in your usual classroom.
MLRC problem session: Wednesday May 10: 4:15 slot
Office visits: Tuesday 7:45-4, Wednesday 7:45-4, Thurday 7:45-4 (approximately).