Math 20d Winter 2008
Homework Assignments
Warmup Homework (Do not hand in!)
Do these problems by January 11
Section 1.1:
2, 6, 13, 18, 19, 20, 23
Section 1.3:
1, 3, 5, 8, 11, 17
Homework 1
Due by 5pm on January 18 in boxes on 6th floor of AP&M.
Section 2.1:
15, 20, 22(bc), 30, 31.
Section 2.2:
2, 4, 7, 14, 30(abcde), 32(ab).
Section 2.3:
1, 2, 7, 9, 16, 17.
Homework 2
Due by 5pm on January 25 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
Section 2.4:
7, 14, 22. Extra practice: 8, 13.
Section 2.5:
3, 15, 20, 22. Extra practice: 10, 14, 21, 23.
Section 2.6:
1, 8, 10, 16. Extra practice: 3, 7, 9, 15.
Section 3.1:
1, 4, 9, 12, 18, 23. Extra practice: 3, 10, 11, 24.
Homework 3
Due by 5pm on February 1 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
IMPORTANT: Work through the problems for Section 3.2 and 3.3 before Exam 1. Exam 1 will cover all of the material we have covered up through the lecture of Wednesday January 23; this is all sections up through and including Section 3.3 (though we only covered a bit of Section 3.3).
Section 3.2:
5, 14. Extra practice: 2, 4, 15.
Section 3.3:
1, 4. Extra practice: 2, 6.
Section 3.4:
8, 12, 15, 19, 21, 27. Extra practice: 3, 4, 18, 22.
Section 3.5:
2, 3, 6, 11, 13, 23. Extra practice: 1, 9, 12, 25.
Homework 4
Due by 5pm on February 8 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
IMPORTANT: Before 2:15pm on Monday 2/4/08, the problems from Section 3.6 below were incorrectly labeled as Section 3.7. There is no homework from Section 3.7 for this week. I am very sorry for the inconvenience. The problems listed below are now correct.
Section 3.6:
2, 3, 5, 7, 12, 14. Extra practice: 4, 5, 8, 11, 13.
Section 3.8:
1, 3, 13, plus problems below. Extra practice: 2, 4
In addition, do the following three problems and hand in as part of your homework. The setup for all of them is the same: consider a mass on a spring with dashpot (damping): the object has mass m, the damping constant is gamma, and the spring constant is k, so the equation for the position function x(t) is mx'' + gamma x' + k x = 0. Also the initial position is x(0) = x
0
and the initial velocity is x'(0) = v
0
.
For each of the following sets of values, find a formula for x(t); if both sine and cosine appear in the answer, reexpress the answer using cosine only (as in the problems 3.8#1,3 above); sketch the graph of the solution (as carefully as possible); determine if the system is undamped, overdamped, critically damped or underdamped; and finally briefly describe the motion of the mass in words.
A. m = 1/2, gamma = 3, k = 4, x
0
= 2, v
0
= 0.
B. m = 1, gamma = 8, k = 16, x
0
= 5, v
0
= -10.
C. m = 1, gamma = 10, k = 125, x
0
= 6, v
0
= 50.
Solutions:
A: x(t) = 4e
-2t
- 2 e
-4t
B: x(t) = 5e
-4t
+ 10te
-4t
C: x(t) = e
-5t
(6 cos 10t + 8 sin 10t) (I leave it to you to express the answer in the form Re
-5t
cos(10t - delta).)
Homework 5
Due by 5pm on February 15 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
Section 3.7:
3, 7. Extra practice:1, 2, 5.
Section 7.1:
1, 5. Extra practice: 2, 6.
Section 7.2:
1, 22. Extra practice: 2, 6, 23.
Section 7.3:
16, 17, 19, 20. Extra practice: 15, 18.
In addition, do the following two problems and hand in as part of your homework. The setup for all of them is the same: consider a mass on a spring with no damping: the object has mass m, and the spring constant is k. There is also an external force F(t) applied to the system (dependent on time). Thus the equation for the position function x(t) is mx'' + kx = F(t). Assume the initial position is x(0) = x
0
and the initial velocity is x'(0) = v
0
.
For each of the following sets of values, find a formula for x(t), sketch the graph of the solution, and finally briefly describe the motion of the mass in words.
A. m = 1, k = 4, F(t) = cos t, x
0
= 1/3, v
0
= 0.
B. m = 1, k = 4, F(t) = -4 sin 2t, x
0
= 0, v
0
= 1.
(The phenomenon you will notice in problem B is called
resonance
. It can occur in systems which are being externally driven at a frequency which matches a natural frequency of vibration of the system---this matching manifests itself mathematically as the problem of "duplication" in the method of undetermined coefficients. Resonance is the process that causes, for example, a glass to shatter when sounds of a certain frequency are applied: the amplitude of the resulting vibrations gets bigger and bigger. The avoidance of resonance is also an important consideration in bridge design, since resonance has led to bridge collapse in several famous incidents.)
Solutions:
A: x(t) = (1/3) cos t.
B: x(t) = t cos 2t.
Homework 6
Due by 5pm on February 22 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
Section 7.5:
4, 6, 8, 15. Extra practice: 1, 5, 6, 16.
Section 7.6:
3, 4, 10, 13. Extra practice: 1, 5, 9.
In all of the problems that ask you to draw something, I want you to draw a
phase portrait
, that is a (rough) sketch of some trajectories with arrows indicating the direction of motion for positive t. Follow the examples in class. If the problem ask you to plot a direction field first, you do
not
have to do that.
Homework 7
Due by 5pm on February 29 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
IMPORTANT: Exam 2 will cover all of the material we have covered in Sections 3.1-7.6. (Although sections 3.1-3.3 were also part of Exam 1, all of Chapter 3 is interrelated and so the material from those sections is still relevant.) The material in the homework problems below will not be on the exam.
Section 7.7:
3, 11, plus additional problem below. Extra practice: 4, 6, 12.
Section 7.8:
2, 3, 4. Extra practice: 1, 9.
In addition, do the following problem and hand in as part of your homework.
Consider the following 2 by 2 matrix A:
( 1 1 )
( 0 1 )
Caclulate the matrix exponential e
At
, directly from the power series definition. (The first step is to find a formula for A
n
for each n. Calculate it for a few small n until you see the pattern. The power series you get in the top right coordinate may take a little work to see that it is a familiar function.)
Then, calculate the fundamental matrix Phi for the system x' = Ax by the more direct method we learned first in class. The answer you get for Phi should be the same matrix e
At
you calculated earlier.
Other comments:
In 7.7#3, find the fundamental matrix Phi satisfying Phi(0) = I only; you don't have to find another fundamental matrix first.
In 7.8#2,3,4 I want you to draw a phase portrait, following the examples in class. Do not draw a direction field first.
Homework 8
Due by 5pm on March 7 in boxes on 6th floor of AP&M. The problems marked "Extra practice" are not to be handed in, but you may find doing them helpful.
Section 5.2:
7, 9. Extra practice: 1,2.
Section 5.3:
5, 6, 7. Extra practice: 8.
Section 6.1:
5(a), 11. Extra practice: 7, 12.
Section 6.2:
5, 8, 12, 14, 16. Extra practice: 2, 4, 7, 11, 13, 15.
Hint for 5.3#7: You need to determine the complex solutions of x
3
= -1. These are the three complex cube roots of -1. You learned how to find these in 20b! Also, you need to figure out the distance in the complex plane between these roots and some other numbers. Remember that a + bi is plotted as the point (a,b) in the complex plane. So the distance between two complex numbers (a + bi) and (c + di) is the same as the distance between the points (a,b) and (c,d) in the plane.
Homework 9
Do not turn this homework in! We are not going to grade it and it will not be scored. However, doing these problems will be crucial in preparation for the final exam.
Section 6.2:
21, 23. Extra practice:20, 22.
Section 6.3:
1, 5, 7, 9, 14, 15 Extra practice:2, 4, 8, 11, 16, 17.
Section 6.4:
1, 2, 5. Extra practice: 6, 7, 9.
Section 6.5:
1, 2, 7. Extra practice: 3, 5, 8.
Department of Mathematics
Math 20d