This is the last assignment for the quarter and it is nominally due on Friday 12/5. However, I know that several students have other things due on that date, so you can turn your assignment in on the morning of Monday 12/8 without penalty and we plan to post solutions Monday afternoon.

Section 3.3 # 1, 4, 10 (Also convert #1 to an equivalent system of 1st order equations. Solve it in that form and compare with your previous answer.)

Solve the following systems of linear 1st order ODEs:
(Note that the independent variable is t.)

a)
x' =   x -  y + 4z
y' = 3x + 2y - z
z' = 2x +  y  - z  +  t

b)
x' = 3x - 3y + z
y' = 2x -  y
z' =   x  - y + z
x(0)=7, y(0)=4, z(0)=3

Section 11.1 # 5

Section 11.2 # 3, 10

Section 11.3 # 1, 13 and find a particular solution of y'' + 4y = f(x) where f(x) is the function in problem 13. Plot the sum of the first 3 or 4 terms in your solution. Is the solution periodic? Why or why not?

Due Wed. 26. (Anyone who wants to work on this over the Thanksgiving holiday is welcome to turn it in on Monday Dec. 1. Rember that there is no class on Friday Nov. 28.)

Index Notation Problems: Use index notation to verify the pairs of equations labeled as (14) and (16) in the Summary of Ch. 9 on p. 419.

ODE Problems: Section 2.1 # 22 (Reduction of order)

Section 2.2 #29, 30, 33 (Homogeneous linear 2nd order ODEs)

Section 2.4 # 3 Do the problem as written, then describe how the solution changes if you double the intial displacement. Now do some numerical simulations of the original (not linearized) pendulum equations. In that case, what happens when you double the initial displacement? What happens to the period of the oscillation as the displacement becomes large? (Modeling and nonlinearity)

Section 2.5 # 11, 12 (Euler-Cauchy Equations - variable coefficients)

Section 2.7 # 5, 6, 18 (Nonhomogeneous 2nd order ODEs)

Section 2.10 # 3, 8

Due Wed. Nov. 12. (Note that this material will be covered on Exam #2 next Friday Nov. 14, so solutions will be posted late on the 12th to allow for exam preparation. Everyone please get your assignments submitted by Wed afternoon.)

Section 10.1 # 1, 3

Section 10.2 # 2, 7, 17, 18

Section 10.3: Review on your own as needed

Section 10.4 # 7, 8, 9

Section 10.7 # 17, 18 (note that the volume of a cone is (1/3)*(Area of base) * height )

Section 10.8 # 9

Section 10.9 # 11, 13, 18

Due Wednesday Oct. 15 (EDGE assignments should be submitted by 2PM on Oct. 17.)

1. Consider the following sets of vectors. Use the Gram-Schmidt orthogonalization process to obtain an orthonormal basis for the space that they span and determine the dimension of that space. (Note that I am using row vector format to make text entry easier.) Use Mathematica (or some computing resource) to check your result. (If you feel so inclined, you might try creating graphics corresponding to random linear combinations as a way of visualizing the span...)

a) ( {1,0}, {1,1}, {2,1} )

b) ( {1,0,-1}, {1,1,1}, {3,2,1}, {3,4,5} )

Problems from the text:

Section 7.2 # 1, 21 (Matrix operations)
Section 7.3 # 5, 9, 16 (Gaussian elimination)
Section 7.4 #7, 12, 23 (Rank and dimension)

NOTE: The problems below (in Section 7.7) are postponed to become part of HW#4!!!
Section 7.7 # 18, 19, 24a,b (Cramer's rule. Again, use your computing resources to verify your results.)

Due in class on Wed. Oct. 8 (As usual, EDGE students, please make sure your materials are submitted before Friday at 2PM so that solutions can be posted in a timely fashion.)

Do the computing problems at the end of the "Introduction to Mathematica" notebook available on the donwload page.

In the text:
Section 7.9 # 1, 5, 9

The problems below relate to the vectors:
u = {1,1,1}; v = {1,2,3} ; w = {3,4,12}

10. Compute the length of each vector.

11. Are any of them parallel or pendicular to one another?

12. Create a unit vector parallel to each of the vectors.

13. Compute the projection of each vector onto the direction associated with the other vectors.

14. Does the set of linear combinations of these vectors form a vector space? Why or why not? If so, what is the dimension of the space? Give a basis for the space.

15. Pick a random vector of length 3 and compute its projection onto u, v, and w. If you sum these projections, do you recover the vector you originally chose? Why or why not?How would you change your choice of vectors to avoid this "difficulty".