## MATH 22B (SECTION 1), 1204 Haring, 9-10 MWF

Instructor: D. A. Kouba

Office: 572 Kerr

Phone: (530) 752-1083

Office Hours: 10-11 T Th or by appointment

TA Office Hours : John Hong, 2-4 MW, 479 Kerr

EXAM DATES:
• EXAM 1-- MONDAY, JANUARY 26

• EXAM 2-- FRIDAY, FEBRUARY 13

• EXAM 3-- MONDAY, MARCH 9

• FINAL EXAM -- SATURDAY, MARCH 21
The course will likely cover the following sections in our textbook (Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima) -- 1.1, 2.1-2.3, 2.5, 2.7, 2.8, 3.1-3.7, 7.1-7.9, 6.1-6.3

SOLUTIONS TO ALL HOMEWORK ASSIGNMENTS CAN BE PURCHASED AT CLASSICAL NOTES IN THE MU. In addition, you may look at a copy of solutions during office hours in 572 Kerr.

The following homework assignments are subject to minor changes. An asterisk (*) indicates those problems to be handed in.

• HW # 1 ..... p. 10: 1-3, *4, 5, *6, 7, *8, 9, *10, 11, 12, *13, 14, 17, *18, 20, *29 (Use y=-3/2, -1, -1/2, 0, 1/2, 1, 3/2), 31

• HW # 2 ..... p. 23: *2c, *4c, 8c, 9c, 14, *15, *18, 20, *22, 23, *28 and Direction Fields Problem

• HW # 3 ..... p. 30: 2, *5, *8, *9, *11, 12, *16, *20, *21, 22, *23, *26, 29, 35 and *Handout 1

• HW # 4 ..... p. 30: 14, *37b, *38 ..... and ..... p. 38: *2, *3, *5, *6, *11a, *13a, 14a, *18a, *19a (tricky), 20a, *23 (relative minimim)
HINT for problem 37b on page 30 : Let v = y^(1-n) then dv/dt = (1-n)y^(-n) dy/dt. Now multiply the original D. E. by (1-n) y^(-n), simplify, and you should be left with a 1st-order linear D.E. in the variable v !

HINT for problem 19a on page 38 : Recall that sin A = B and A = arcsin B are the same only if A is between -pi/2 and +pi/2 ! If A is in the 3rd or 4th quadrant then sin A = B leads to A = pi - arcsin B !!!!

• HW # 5 ..... p. 79: 1, *2, *4, *6 ..... and ..... p. 38: 29, *30 ..... and ..... p. 54: 1, *22
HINT for problem 29 on page 38 : Use polynomial division first to integrate variable y !
• HW # 6 ..... p. 88: *(1-12), 15, *16, 18, *19, 21, *22 ..... and ..... p. 79: 13a, *14a ..... and ..... p. 54: 2, *24

• EXAM 1 is Monday, January 26, 1998. It will cover handouts and examples from class, homework assignments 1 through 6, and material from sections 1.1, 2.1, 2.2, 2.3, 2.5 (problems assigned only), 2.7, and 2.8 in the book which was presented in lecture notes through Wednesday, January 21, 1998. Most of the exam questions will be homework-type questions. In addition, know the proof for testing whether a differential equation is exact or not. Expect one NEW type of question on the exam where you will be asked to solve a D. E. by using a suggested substitution, as was done on several homework problems. There will be one OPTIONAL EXTRA CREDIT problem on the exam.

SOLUTIONS TO EXAM 1 CAN BE PURCHASED AT CLASSICAL NOTES IN THE MU. The grading scale for Exam 1 was

100-106 A+
85-99 A
70-84 B
58-69 C
47-57 D
0-46 F

• HW # 7 ..... p. 88: *25, 26, 27, *29 ..... and ..... p. 128: 2, *3, *5, *8, *9, *12, 16, *18, *21, *29, 32

• HW # 8 ..... p. 138: 2, *4, 5, *6, 9, *11, 12, 13, *14, 15, *17, *19, *21, 22, 23, *25 ..... and ..... p. 129:*33 ..... and ..... p. 54: *3 (Assume that the rate of decay is proportional to the amount present.), 6

• HW # 9 ..... p. 144: *1, *2, 4, *5, *6, *7, *10, *12, 13, *15, 16, *19, *20, 22, *23, 28 ..... and ..... p. 138: 28, *31 ..... and ..... p. 54: 15, *26

• HW # 10 ..... p. 150: 2, 5, *7, *9, *11, 15, *17, *18, *19, 25, *27 ..... and ..... p. 144: *27 ..... and ..... p. 54: *27
HINT for problem 27 on page 54 : Let V be the volume (ft.^3) of CO in the room at time t minutes. Then set up a differential equation in the fashion of the salt and water mixture problems by comparing the volume (ft.^3) of CO to the volume (ft.^3) of air and multiply by 0.1 ft.^3 per minute to get the proper units.
• HW # 11 ..... p. 159: *1, 3, *4, 5, *9, *12, *14, *23, *25, *26, 27 ..... and ..... p. 150: *39, 40

HINT for problems 39 and 40 on page 150 : The substitution x = ln t together with the chain rule convert the differential equation t^2 y'' + A t y' + B y = 0 (with t as the independent variable) into the differential equation d^2 y/dx^2 + (A-1) dy/dx + B y = 0 (with x as the independent variable). This will be shown in my solution set for HW #11. You can simply use this fact to do these problems.
• HW # 12 ..... p. 171: *1, *2, *3, *4,*5, 6, *7, 8, 14, *16, 19a, *20a ..... and ..... p. 159: 19, 28, *41, 42 ..... and ..... p. 150: *41, *42

• HW # 13 ..... p. 177: *3, *5, 6, 7, 8, *9, *10, 13, *17 ..... and ..... p. 171: 23, *24

• EXAM 2 is Friday, February 13, 1998. It will cover handouts and examples from class, homework assignments 7 through 13, and material from sections 2.5 (problems assigned only), 2.8 (finding integrating factors for nonexact differential equations), 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7. You must know how to solve Euler Equations (problems 39-42 on page 153) using the shortcut given in solution set #11 and given in the HINT above. In addition you must know the PROOFS for Theorems 3.2.2, 3.2.3, 3.3.1, and 3.3.2. You will have to prove ONE of them on the exam. Most of the exam questions will be homework-type questions. There will be eleven (11) problems (3 characteristic equation, 1 exact, 1 undetermined coefficients, 1 variation of parameters, 1 Euler, 1 proof, 1 linearly independent/dependent, and 2 others) and one OPTIONAL EXTRA CREDIT problem on the exam.

SOLUTIONS TO EXAM 2 CAN BE PURCHASED AT CLASSICAL NOTES IN THE MU. The grading scale for Exam 2 was

100-104 A+
86-99 A
70-85 B
57-69 C
49-56 D
0-48 F

• HW # 14 ..... p. 340: { See book for help on the next three: *1, 4, 5 } , { For the next three solve only for x1 and x2: *7, 8, *11 } , 14, *15, 16 ..... and *Q1 { Compute r to 4 decimal places, and c1 and c2 to 2 decimal places } and Q2 from class

• HW # 15 ..... p. 351: 1bc, *2ad, 3b, 4, 6, *9, *10, 11, *13, *14, 22, *23, *24, 26 ..... and ..... p. 340: *21ab (and solve system of D. E.'s )

• HW # 16 ..... p. 363: *1, *2, *3, 4, 5, *6, *7, 8, 9, *12 ( Divide each row of the matrix by e^(-t). ), 13 ( Let t=pi/2 in the matrix. ), *15, *16, 17, 18, *22

• HW # 17 ..... p. 369: 6ab, *7ab ..... and ..... p. 363: *14, 20, *21, 23 ..... and ..... p. 54: *16 ( Find growth constant, assuming NO predation. Then use this constant when setting up a differential equation, assuming predation. )

• HW # 18 ..... p. 378: { Find general solution only on following six problems. On Problems 1 and 11 also verify that solutions are linearly independent: *1, *4, 7, 9, *11, 14 } , *19, *20, 22 ..... and ..... p. 54: *19 ( See Newton's Law of Cooling on p. 53. )

• HW # 19 ..... p. 387: { Find solution only for first four problems: 2, 3, 7, 9 } , 21 (Use the fact that t^(ci) = e^( (ln t ) ci ) = e^( (c ln t) i) and then use Euler's Formula.) , 22 ..... and ..... p. 378: 25 (DIRECTIONS: For the initial value (2, 3) sketch x1, x2, and their trajectory and find a 2 by 2 matrix A for which these functions solve X' = AX.) ..... and ..... p. 54: 21

• HW # 20 ..... p. 396: (Find solutions only on this page.) *1, 3, *5, *6, *13, { For the following two problems assume that the second solution has the form X = (xi) (t^r) ln t + (eta) (t^r), which leads to the equation (A-rI) (eta) = (xi) : *14, 15 } ..... and ..... p. 378: 27 (DIRECTIONS: For the initial value (2, 3) sketch x1, x2, and their trajectory and find a 2 by 2 matrix A for which these functions solve X' = AX.)

• EXAM 3 is Monday, March 9, 1998. It will cover handouts and examples from class, homework assignments 14 through 20 (and problem 16 on p. 396 from HW # 21), and material from sections 2.5 (problems assigned only), 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, and 7.7. Most of the exam questions will be homework-type questions. You must know how to solve systems of the form t X' = AX. There will be 7 problems plus one OPTIONAL EXTRA CREDIT problem on the exam.

SOLUTIONS TO EXAM 3 CAN BE PURCHASED AT CLASSICAL NOTES IN THE MU. The grading scale for Exam 3 was

100-110 A+
85-99 A
70-84 B
60-69 C
50-59 D
0-49 F

• HW # 21 ..... p. 404: *1, *5, 6, *8, *10 ..... and ..... p. 396: 16

• HW # 22 ..... Use Undetermined Coefficients -- p. 411: *7, *4, *8, *1, 14

• HW # 23 ..... Use Variation of Parameters -- p. 411: *5, *15, (Do not simplify the part. sol. Xp on next two problems: 3, 11) ..... and ..... Use Undetermined Coefficients on *P1.) and P2.) from class.

• HW # 24 ..... p. 294: *1, *5ab, *11 (Use the definition of Laplace transform), *15

• HW # 25 ..... p. 303: *1, 2, *3, 5, *8, *11, 13, 21, *22, *23

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OFFICE HOURS For The Week Of March 16-21, 1998
• Kouba in 572 Kerr -- MON. 10-11 and 1-2
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Kouba MAY be out of town on University business from Tuesday, March, 17, through Friday, March 20, 1998. For assistance please see office hours below.

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• John in 479 Kerr -- MON. 2-4, WED. 2-4, THURS. 2-3, FRI. 1-2

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The Math 22B FINAL EXAM is Saturday, March 21, 1998, 8-10 am in 1204 Haring

The final exam will have 14 problems. There will be two (2) systems of differential equations to solve, two (2) applications, five (5) differential equations solved by the methods of your choice, and five (5) OTHER problems; in addition, there will be 3 OPTIONAL extra credit problems. Most problems will be homework type of problems, like those on the three hour exams. All those topics and concepts that you were asked to learn for the three hour exams, you must know for the final exam. There is one major EXCEPTION. There will be NO PROOFS on the final exam. When solving differential equations, I may tell you to use a certain method or I may leave it for you to decide which method to use. You should have a graphing calculator at your disposal, since you may be asked to graph solutions or trajectories. Expect 3 or four problems on the material covered since the third hour exam. You need to MEMORIZE the following Laplace transform formulas-- on page 300, formula numbers 1, 2, 3, 5, 6, 9, 10, 11; and Theorem 6.2.1 and Corollary 6.2.2.

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