M340 – Section 005: Spring 2008
Lectures
HOMEWORK DIRECTIONS: Please write only on one side of each page and show all your work clearly and legibly. I will post the subset of problems from these sections that the graded problems will be chosen from the Friday before the assignment is due, but you should hand in ALL problems. For each section you will receive 10 points for completeness and three problems from each section (chosen from the subset posted on a Friday) will be graded with partial credit up to 5 points each. Therefore, for each section of homework in an assignment, you can receive up to 25 points.
For example, the first homework assignment has 4 sections, so the assignment is worth 100 points. The problems are in the lecture notes and for section 2.1, the problems to be handed in are 1, 2, 6-8, 11, 15-19, 24, and 38. I might choose problems 8, 11, 16, 18, and 24 as possible candidates for grading, and will grade three of these problems.
ADDITIONAL HOMEWORK DIRECTIONS:
Homework #1: Problems from sections 2.1, 2.7, 2.2, and 2.4 are due on Monday, February 4th. See the lecture notes (lectures 02 – 05) for problems.
Possible graded problems:
Homework #1 HINT: Section 2.2 #42 – Use the model derived in problem 24(b) in section 2.7. The only differences between this problem and problem 24(b) of section 2.7 is that the cross-sectional area “A” is not constant in this problem (it is a function of y, you need to find this function A(y)), and the height variable h is now replaced with the height variable y. Leave your answer in terms of “a” and “g.”
Graded Problems:
Homework #2: Problems from sections 2.8, 2.9, and 2.3, 2.5 (lectures 06-08) are due on Monday, February 11th.
Possible graded problems:
Graded problems:
Homework #3: Problems from sections 3.1, 3.3, 6.1, and 6.2 (lectures 09-11) are due on Monday, February 18th.
Graded Problems:
Homework #4: Problems from sections 7.1, 7.3, and 7.4 are due on Monday, February 25th.
Graded Problems:
Homework #5: Problems from sections 7.5, 7.6, 7.7, and 8.1 are due on Monday, March 3rd.
Graded Problems:
Homework #6: Problems from sections 8.2, 8.3, 8.4, 8.5 are due on Friday, March 14th.
Graded Problems:
This assignment will have several problems graded for feedback, but the grade will be determined on completeness, so enjoy your break.
Homework #7: Problems from section 9.1 are due on Monday, March 24th.
Graded Problems:
Section 9.1 graded problems: 22, 24, 44, 48, 54 (Recall from class that in Matlab “>> [V,D]=eig(A)” returns the V and D matrices shown in problem 53, you can use Matlab to obtain V and D for 54 and 55 in your homework and just use hand calculations to work out the matrix multiplication. You should obtain the inverse of V using hand calculations by augmenting with the identity matrix and putting the augmented matrix into reduced row echelon form)
Homework #8: Problems from sections 9.2 and 9.5 are due on Monday, March 31st.
Graded Problems:
Homework #9: Problems from section 9.6 are due on Friday, April 4th.
Graded Problems:
Hint: For #13 you might find it useful to first look at equations (6.18)-(6.20) on p.421 of your text. Then use equation (6.8) on p.417 for one part of the equation, and use the series definition of the exponential of a matrix for the other part. Then you’ll use the fact that if an n-by-n matrix B has a nullspace with dimension n, then the matrix B is the zero matrix (for this problem n=2 and B =(A-lambda*I)^2, and the fact that B has nullspace with dimension 2 follows from Theorem 5 of lecture 23 (also restated as Theorem 9 of the review material handed out on Monday)).
To use ode45 for the system I did in class (for section 9.7 problem 9):
The matrix A had as its first row [-3, -4, 2], as its second
row [-2, -7, 4], and as its third row [-3, -8, 4], so to use ode45 to solve it,
I first define a function in the editor in Matlab
named prob as follows:
function xd=prob(t,x);
xd=zeros(3,1);
xd(1)=-3*x(1)-4*x(2)+2*x(3);
xd(2)=-2*x(1)-7*x(2)+4*x(3);
xd(3)=-3*x(1)-8*x(2)+4*x(3);
end
Then in the command prompt of Matlab I type:
[t,x]=ode45(@prob,[0
8],[10; pi; 1]);
where [0 8] is the time interval I want it to solve the problem on (this is up
to you to choose for your problems), and [10; pi; 1] was my arbitrary initial
condition for the system, which is again up to your discretion to choose. The
reason it wouldn't run in class was I didn't initialize the xd as a vector in the function. If the dimension of
the problem is "d" then the first thing you should type in your
function is xd=zeros(d,1);
to initialize a dx1 vector. My advice is to copy and paste the function above
into the Matlab editor and then adjust the
coefficients and dimension of the problem as necessary. After obtaining x, you
want to plot the 3 components of x (the “x”, “y”, and “z” components) by typing
plot3(x(:,1),x(:,2),x(:,3));
For problems 17-19, use the eig command to get the V and D matrices, and then use the hold command and the plot3 command to plot all the exponential solutions using +/- the vectors from V times the associated exponential of the eigenvalue times t, with
t=-10:0.01:10;
and let’s say the first column in V is [2; 3; 1] (read downwards as the first column of V), and the first diagonal of D is 7, then you’d let
y=[exp(7*t)*2, exp(7*t)*3, exp(7*t)*1];
and then you’d type
plot3(y(1,:),y(2,:),y(3,:)); hold;
plot3(-y(1,:),-y(2,:),-y(3,:))
and let’s say that the second column in V is [1;0;1], and the second diagonal of D is -2, then
y=[exp(2*t)*1, 0, exp(2*t)*1];
plot3(y(1,:),y(2,:),y(3,:));
plot3(-y(1,:),-y(2,:),-y(3,:))
and then repeat for the third exponential solution taken from the third column of V and the third diagonal entry of D. Then choose 8 initial conditions and let ode45 solve the trajectories and plot them on the same axes as these solutions to finish the problem.
Homework #10: Problems from sections 9.3, 9.4 and 9.7 are due on Tuesday, April 8th.
Graded Problems:
TBD
For 9.9 when you use the variation of parameters technique on a system larger than 2 dimensions, you should use Matlab and the expm command.
>> syms t %This defines a symbolic variable t in Matlab
>> expm(A*t) %This will give you the matrix exponential (you have to have entered A earlier)
>> expm(-A*t) %This gives the matrix exponential inverse
You can then finish the problem by hand once you have expm(A*t) and exp(-A*t) if you look in the notes for the variation of parameters formula that uses the matrix exponential instead of the fundamental matrix. Let me know if you have any other questions.
Homework #11: Problems from section 9.9 are due on Friday, April 11th.
Graded Problems:
TBD
Homework #12: Problems from sections 4.3 through 4.7 and 9.8 are due on Wednesday, April 23rd.
Graded Problems:
TBD
Homework #13: Problems from sections 5.1, 5.2, 5.3, and 5.4 are due on Friday, May 2nd.
Graded Problems:
TBD
Homework #14: Problems from sections 5.5, 5.6, and 5.7 are due on Friday, May 9th.
Graded Problems:
TBD
NO MORE HOMEWORK AFTER HOMEWORK#14!!!! WOOOOOO!!!! HOWARD DEAN SAYS “YEEEAAAAA!!!!” Thanks for the great semester. I am really proud of you guys/gals for all your hard work. I don’t think I’ve ever been more demanding of work in any class and you all have done a fantastic job of working hard and with determination. It has been my pleasure to be your instructor. I feel really privileged to have had you all as my students this semester. You should be very proud of yourselves. You have done a great job.
Tentative Weekly Schedule
|
Week # |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
|
1 |
|
1.1-1.3 (01) |
2.1 (02) |
|
2.7 (03) |
|
2 |
2.2 (04) |
2.4 (05) |
2.7 (03) |
|
2.8 (06) |
|
3 |
2.9 (07) |
2.3, 2.5 (08) |
Matlab lab (meet in Weber 205) |
|
3.1, 3.3 (09) |
|
4 |
6.1 (10) |
6.2 (11) |
Matlab lab (meet in Weber 205) |
|
7.1 (12) |
|
5 |
7.2, 7.3 (13) |
7.2, 7.3 (13) |
7.4 (14) |
|
7.5 (15) |
|
6 |
7.6 (16) |
7.7 (17) |
8.1 (18) |
|
8.2, 8.3 (19) |
|
7 |
Review (regular room, E203) |
Review (regular room, E203) |
Review (regular room, E203) |
TEST |
|
|
8 |
8.4, 8.5 (20) |
Finish 8.4, 8.5 (20), start 9.1 (21) |
Finish 9.1 (21) |
|
Start 9.2, 9.5 (22) |
|
9 |
9.2, 9.5 (22) |
9.6 (23) |
9.6 (23)/Matlab lab (meet in Weber 205) |
|
9.3, 9.4 (24) |
|
10 |
Review of 9.1, 9.2, 9.5, 9.6 |
9.7 (25) |
9.9 (26) |
|
9.9 (26) |
|
11 |
9.9 (26) maybe 9.8, 4.3 (27) |
9.8, 4.3 (27) |
4.4 (28) |
|
4.5, 4.6 (29) |
|
12 |
4.7 (30) |
Review (regular room, E203) |
Review (regular room, E203) |
TEST |
5.1, 5.2 (31) |
|
13 |
5.1, 5.2 (31) |
5.3, 5.4 (32) |
5.3, 5.4 (32) |
|
5.4 (32) |
|
14 |
5.5 (33) |
5.6 (34) |
5.7 (35) |
|
5.7 (35) |
|
15 |
Review/Finish Chp.5 |
Review/Finish Chp.5 |
Review |
|
Review |
Office Hours: Monday, Tuesday, Wednesday 5-6, or by appointment
Programming Euler’s method in Matlab