MAP 2302 homework

MAP 2302 homework Homework Assignments
MAP 2302, Sections 3227 & 5010 — Elementary Differential Equations
Fall 2007

Last update made by D. Groisser Sat Dec 15 21:34:08 EST 2007

Homework problems and due dates (not the dates the problems are assigned) are listed below. This list, especially the due dates, will be updated frequently, usually in the late afternoon or evening the day of class or the next morning. Due dates and assignments more than one lecture ahead are estimates; in particular, due dates may be moved either forward or back, and problems not currently on the list from a given may be added later (but prior to their due dates, of course). On a given day there may be problems due from more than one section of the book.

Exam-dates and some miscellaneous items may also appear below.

If one day’s assignment seems lighter than average, it’s a good idea to read ahead and start doing the next assignment, which may be longer than average.

Unless otherwise indicated, problems are from our textbook (Nagle, Saff, & Snider, 4th edition). It is intentional that some of the problems assigned do not have answers in the back of the book or solutions in a manual. An important part of learning mathematics is learning how to figure out by yourself whether your answers are correct.

Read the corresponding section of the book before working the problems. Don’t read just the examples, and don’t just try the homework problems and refer to the text only if you get stuck.

Date due	Section # / problem #’s
M 8/27/07	Read the syllabus and the web handouts “Taking notes in a college math class” and “What is a solution?”, obtainable from the Web. Read Section 1.1 and do problems 1.1/ 1-16. We did not get to “order” or “linear vs. non-linear” in class on Wednesday, but you should have no trouble doing these problems based on pp. 4-5 of the book, and I will review this material on Monday. Also, do non-book problem #1. Note: if you have a problem viewing any of my PDF files, check your version of Adobe Acrobat at http://www.adobe.com/products/acrobat/ . You may need to update (all you need is a recent version of the free Adobe Reader).
W 8/29/07	1.2/ 1, 3-5, 14, 15, 19.
F 9/1/07	1.2/ 2, 10, 11, 30. Do non-book problem #2. Read pp. 12-14. Read sections 2.1 and 2.2. In your reading, make sure you understand the discussion following Example 3 (bottom of p. 44 to top of p. 45).
W 9/5/07	1.2/18, 23-29. 2.2/1-5, 8, 9, 11, 13, 17, 18, 23, 28-31. For problems in the 7-16 group and the 17-26 group, use the method that I used in class for the example “dy/dx = xy“. However, remember that “Solve the equation” means “Find the general solution of the equation” (i.e. find all solutions). For some of the problems in the 7-16 group, the method that I used in the above example is not sufficient to find all the solutions, so you may not be able to finish these problems completely yet. Problem 30 of this assignment illustrates the relevant issue, but on Wednesday we will see how to find systematically the solutions that are missed by separating variables. Save your work on this assignment so that in the next assignment, you can fill in any solutions you missed the first time through.
Every Friday	Homework discussion: Starting 9/7/06, I will hold a weekly session on Friday, 9th period, in Little 201, for the purposes of answering homework questions for recently assigned homework–problems that were due that week, or, if a week was skipped, since the last homework-discussion. Attendance is optional; however I will continue holding this session only as long as at least 10 students show up (out of the 63 in my two sections put together, not 10 for each section); I will discontinue it the second time that fewer than 10 show up. I stress again that I expect you to be doing all your homework by the due-dates, and that these homework sessions are intended for recent homework problems, not to help you catch up the week before an exam. If I cancel the sessions because there’s insufficient interest (i.e. fewer than 10 attendees) when an exam is not imminent, I will not re-institute them the week before an exam.
F 9/7/07	Re-examine your work on problems 2.2/8, 9, 11, 13, 17, 18, 23. See if you really found all solutions; add to your final answers any solutions that you previously missed. 2.2/27abc, 32-34. Read Section 2.3.
M 9/10/07	2.2/19. (This should really have been part of the last assignment.) I’m giving you very little HW this weekend, but there will be a lot due Wednesday and Friday. Budget your time by trying to get ahead in your other courses over the weekend so that you’ll have more time for DE homework Monday and Tuesday, or try to use what you learned from reading Section 2.3 to get a head-start on the HW from that section due Wednesday.
W 9/12/07	2.3/1-6, 7-9, 13-15, 17-20, 22, 23 (read Example 2 on p. 52 first), 24, 27a, 28, 30-33, 35. In #33, note that what you might think is only a minor difference between the DE’s in parts (a) and (b)–a sign-change in just one term–drastically changes the nature of the solutions. When solving differential equations, a tiny algebra slip can make your answers utter garbage. For this reason there is usually no such thing as a “minor algebra error” in solving differential equations. Do non-book problem #3. (This list of problems was updated 9/10/07, so don’t go by what you may have printed out earlier.)
F 9/14/07	2.2/6, 16, 22. In #22, note that although the differential equation doesn’t specify which variable is (in)dependent, the initial condition does. Thus your goal in #22 is to produce an explicit solution “y(x)= …“. Read Section 2.4 through at least the top of p. 62 (the sentence that ends “… do not meet the compatibility conditions.”). If you want to get a head-start on the next assignment or get a preview of what we’ll be doing in class Friday, keep reading through the end of Section 2.4. 2.4/1-8, 27a, 28a. (Use Theorem 2 on p. 61 to decide exactness.) If you read all of Section 2.4, I encourage you to try some the problems that are due Monday.
M 9/17/07	Read the rest of Section 2.4. Read the online handout “A terrible method for solving exact equations”. The parenthetical “we proved it!” on the handout does not yet apply, since I haven’t yet done the proof in class. However, you did read the proof in the book as part of your homework. 2.4/11, 12, 14, 16, 17, 19, 20-22, 32 (note that #22 is the same DE as #16, so you don’t have to solve a new DE; you just have to incorporate the initial condition into your old solution).
EXAM-DATE CHANGE	The first midterm will be given on Fri. Sept. 21.
W 9/19/07	No new homework; start studying for Friday’s exam. If you have done all your homework (and I don’t mean “almost all”), you should be able to do all the review problems on pp. 81-82 except #s 9, 11, 12, 15, 18, 19, 22, 25, 27, 28, 29, 32, 35, 37.
Special note	Here is what I told a 2003 Differential Equations class after they performed less well than they’d hoped on their first midterm. I’m hoping that this advance warning will help you to avoid the flaws in those students’ study habits, exam prep, etc. If you had any unpleasant surprises on [this] exam, please re-read the section on prerequisites on the class home page, and the sections on homework, workload, attendance, and miscellany on the syllabus. You may also want to re-read the handout “Taking notes in a college math class”. Never try to figure out the minimum amount of work you need to do to get the grade you want. Your goal should be to put yourself in a position to get 100% on every exam, regardless of what subset of fair-game material ends up being represented on the exam, and regardless of what form the questions take. If you make this your goal, you will learn and retain far more material, and get far better grades.
F 9/21/07	First midterm exam (assignment is to study for it). The relevant book-sections for the exam are 1.1-1.2 and 2.1-2.4. However, bear in mind that I went into greater depth than the book on certain topics, and there were topics covered in homework that I did not discuss in class. Anything that I covered in class or was covered in the book or in homework is fair game for the exam.
M 9/24/07	Read Sections 4.1 and 4.2.
W 9/26/07	4.1/1-10. Typo correction: In #10a, 2nd line, and in #10d, 2nd line, the function written after “B” should be “sin”, not “cos”.
F 9/28/07	Check that all the operators listed on the table in Wednesday’s class are linear except for the squaring operator. Do non-book problems 4, 6, 7ab.
Grades scales & exam statistics	The grade-scales and some statistics for the first exam are now posted for the 6th-period and 7th-period sections. If you want to see how you fared relative to your classmates, click on the “list of scores” link your section’s page (or the other section’s if you want to compare). On the list-of-scores page, the grouping by first digit effectively gives a histogram of the scores (with 10-point bins), since most scores have two digits and therefore take the same horizontal space.
M 10/1/07	Do non-book problems 5, 7c. Note that there is a paragraph pertaining to problem 7 at the top of p. 2 of the document. Based on your reading (see the HW due 9/24/07) do the following problems: 4.2/1-10, 13, 14, 17, 18, 20, 21, 22, 26. Your answer to 21(b) should be the same as what you would have found by the “integrating factor” method.
W 10/3/07	4.2/ 27-33, 39, 40. Typo correction: #39 has a typo in line 5. The function multiplying c2 should be y3(t). I know that this material looks nothing like what I’ve focused on in class for the last two lectures. As I mentioned, I’m covering important material that used to occupy two sections of the book that were deleted in the last revision. By the end of the next lecture, I should be mostly caught up with what I’ve assigned from Section 4.2. Read Section 4.3.
F 10/5/07	Based on your reading, do problems 4.3/1, 4, 6, 7, 9-12, 17, 18, 28.
M 10/8/07	4.3/ 30, 38, 39. Do non-book problem 8. This, and problems 38-39 in the book, fall into the “semi-easy” category I mentioned in class (DE’s for which a clever change-of-variable(s) turns a nonconstant-coefficient DE into a constant-coefficient DE in the new variable(s)).
W 10/10/05	Complete the steps, started in class, to check that if r is a complex constant, then d(ert)/dt = r ert. Show that if c is a complex constant and h is a complex-valued function, then (ch)’=ch’. Read section 4.5 pp. 186-189 through the end of Example 2, excluding Example 1. We will cover section 4.4 after we have done most or all of section 4.5 , but most of the examples in 4.5 assume (naturally) that you’ve already read 4.4; except for Example 2 they will not make sense to you yet.
F 10/12/07	Read Section 4.4. 4.4/9-11, 14. Hint for #9: 9=9e0x (same idea applies to #10). Hint for #14: 2=eln(2). Based on your reading, try to do as many problems as possible from the assignment due Monday 10/15, so that this material will be less new to you when you study for the midterm.
EXAM-DATE CHANGE	The second midterm will be given on Wed. Oct. 17.
(slight) mistake in textbook	In the colored box on p. 184, insert “and β is not 0” before the period. Similarly, in the colored box on p. 191, insert “and β is not 0” before the last comma. In both cases, “β = 0” puts you in the situation of the top half the box (with α = r), in which case it is possible to have s=2. I don’t like the book’s separation of the “no sine or cosine involved” and “sine and/or cosine involved” cases. Both cases are part of the same master formula. The book’s approach obscures this by putting the “β = 0” case separately, and using a different letter in the exponent when β = 0 (i.e. using r instead of α when β = 0).
M 10/15/07	4.4/1-8, 12, 13, 15-18, 20-24, 27-32 Finish reading Section 4.5. 4.5/9-16, 17, 18, 20, 22, 23-30, 31-36. (READ INSTRUCTIONS–problems 9-16 require no significant computation!) The box on p. 191, “Method of Undetermined Coefficients (Revisited)” covers all the different types of functions g for which L[y]=g can be solved by this method (but see the “(slight) mistake in textbook” warning above).
W 10/17/07	Second midterm exam (assignment is to study for it).
F 10/19/07	Read sections 6.1 and 6.2. (We’re not done with Chapter 4 yet, but this material is relevant to Chapter 4.)
Grades scales & exam statistics	The grade-scales and some statistics for the second exam are now posted for the 6th-period and 7th-period sections.
M 10/22/07	6.1/1-6,29
W 10/24/07	6.2/15-18. 6.2/1-14. I said in class that the only higher-order operators we’ll be using will come to us in factored form. That’s true of the ones I’ll use in class and (probably) on exams, but isn’t true in this group of problems. However, any time you’re able find a root r1 of an nth degree polynomial Pn(r), the quantity r-r1 is a factor of Pn(r)– i.e. Pn(r) = (r-r1) Qn-1(r) for some polynomial Qn-1 of degree n-1–and you can find the quotient polynomial Qn-1(r) by long division of polynomials or by synthetic division, at least one of which you should have learned in high school. When n=3, you can then always factor the quadratic polynomial Q2(r), giving you a complete factorization of P3(r). The author has made problems 1-14 doable by giving polynomials that have an easy root to find. In #1, 0 is a characteristic root. In problems 2-10 and 12, either 1 or -1 is a characteristic root. In problem 11, you may see the factorization fourth-degree characteristic polynomial just from your familiarity with binomial expansions; if not, check whether 1 or -1 is a root (one of them turns out to be), then divide the fourth-degree polyomial by r-1 if 1 is a root or by r+1 if -1 is a root, and repeat the process for third-degree quotient polynomial that you get. In problem 13, you should be able to figure out what to do because only even powers of r appears in the characteristic polynomial. In #14, the author’s hint gives you a pair of non-real conjugate roots, hence two linear factors that you can multiply together to get a real quadratic polynomial. Divide the characteristic polynomial by this polynomial, yielding another quadratic polynomial as quotient, and take it from there.
F 10/26/07	Read Section 6.3. Note that the book’s operator (D-α)2 +β2 is the same as the operator (D-(α+βi))(D-(α-βi)) that I’ve used in class, with α+βi = r1. 6.3/1-4, 11-20, 21-29. Although the book says to use the method of undetermined coefficients for problems 1-4 and the annihilator method for problems 21-29, they are really the same method, so it’s OK to use the method of undetermined coefficients for all these problems. The DE in #29 is third-order, but you should have no problem factoring the characteristic polynomial.
M 10/29/07	Read Section 4.6
W 10/31/07	BOO! 4.6/1-18, 19 (first sentence only), 21, 25.
M 11/5/07	Read sections 7.1 and 7.2. 7.2/21-28.
W 11/7/07	7.2/1-8, 10, 12 (note: “Use Definition 1” means “Use Definition 1”, NOT the box on p. 358 or any other table of Laplace Transforms), 13-20, 29a-d,f,g,j.
EXAM-DATE CHANGE	The third midterm will be given on Fri. Nov. 16. I will still return the graded exams on Mon. Nov. 19 (barring serious illness, etc.), so that students considering dropping the class will have this information in time to drop.
F 11/9/07	7.3/1-8,12,14,19,31 Read Section 7.4. Part of this is a review of partial fractions (pp. 370–373). I will not have time to review this prerequisite Calc 2 material in class, so it is important that you come to class prepared to follow any problem I present that uses partial fractions without asking the sort of questions one would expect from students who have never seen (or have completely forgotten) partial fractions. Read Section 7.5.
W 11/14/07	7.3/9,10,25 7.4/1-10, 15, 16, 20, 21-24, 26, 27. 7.5/1-8,10, 15,21,22,29. To learn some shortcuts, you may want first to read the web handout “Partial fractions and Laplace Transform problems” (pdf file), but I’m not officially assigning this till after the exam.
minor corrections to syllabus	The syllabus that I originally handed out had two minor errors relating to the 7th-period (1:55-2:45) section: (1) near the top of the syllabus, I wrote “8th period” instead of “7th period”; and (2) in the paragraph about the final exam, I wrote the wrong section number (3615) instead of the correct 5010. None of this affects anyone in any material way; the date and time of the final exam are the same as originally written. But if you printed out a copy of the syllabus you may want to fix these mistakes, or view/print the corrected syllabus.
office hours Thurs. Nov. 15	I’ll have office hours, just for my MAP 2302 students, Thursday Nov. 15, 2:00-3:30. I’m making this one-time exception to my no-Thursday-office-hours rule in view of the compressed week and Friday’s exam.
F 11/16/07	Third midterm exam (assignment is to study for it).
M 11/19/07	Read the web handout “Partial fractions and Laplace Transform problems” (pdf file). Read Section 7.6.
Grades scales & exam statistics	The grade-scales and some statistics for the third exam are now posted for the 6th-period and 7th-period sections.
W 11/21/07	There will be class this day. The day before a holiday is not a holiday. 7.6/1-4,6,8,10,11,12,14-18
M 11/26/07	Re-read the material in Section 7.6 on periodic functions (bottom of p. 390 through Example 6 on p. 392). I did not get to go over this in class, but the book’s coverage is straightforward, and you will be responsible for this material. However, you will not be responsible for the portion of Section 7.6 beyond Example 6. 7.6/5,7,9,13,21-28, 29-31, 33, 35-37, 39. For all the problems in which you solve an IVP, write the final answer in “tabular form”. (For those of you who missed class, by “tabular form” I mean an expression like the one given in Example 1, equation (3), for the function f. I.e. do not leave your final answer in the form that’s at the end of Example 1, after you’ve inverse-transformed Y(s). The unit step-functions should be viewed as convenient gadgets to use in intermediate steps to work through a problem efficiently.) Read Section 8.1.
W 11/28/07	Read Section 8.2. This is related to what I covered in class on Monday (all of which was review of material from Calculus 2). Although I’ve already had you read Section 8.1, in class it will be more efficient for me to cover 8.2 first. 8.2/1-6,7,8. Note: “convergence set” in the book is what I initially called “domain of convergence” and, later, “interval of convergence”. In problems 1-6, find only the open interval of convergence; don’t worry about the endpoints.
F 11/30/07	8.2/9,10,11-14,17,19,20,21,22,23,24,27. In all these problems, treat the power series as formal power series; on Friday I will state a theorem that justifies this. 8.1/1,2,4,8-12. (I have not covered this material in class yet, but you should be able to do these problems based on your reading. I will cover some of this material briefly in class on Friday.) Read Sections 8.3 and 8.4.
M 12/3/07	8.2/31,34,37,38 (better hint: integrate the series for 1/(1+x) and figure out what to do next). 8.3/1,3,5,7-10,11-14,18, 20-22,24,25,33. I realize these will be challenging for you to do before I’ve done a single example in class, but I still want you to get started on these right away so you can get as much practice as possible before the final. I will do examples (and hardly any theoretical stuff) on Monday.
W 12/5/07	8.3/32,34. In #34 note that n is not a summation index; it has a different meaning in this problem. This problem can be done without using “Σ-notation”; you shouldn’t need a summation index. 8.4/15, 20,21,23,25, 29 (in #29 be careful not to use the letter n as a summation index since it already has an assigned, different meaning in this problem). Note: When I test you on Section 8.4, I will be more concerned with your knowing how to find power-series solutions than with knowing what Theorem 5 says about their radii of convergence.
Material for final exam	The final will be cumulative, with a disproportionate emphasis on material covered since the last midterm (probably more than 25% but less than 50%). Sections 1.1-1.2, 2.1-2.4, 4.1-4.6, 6.2-6.3, 7.1-7.6 (minus the portion of 7.6 from Example 7 to the end of the section), 8.1-8.3, and the material from 8.4 indicated below. Section 8.4: you are not responsible for the material in this section on radius of convergence. Basically this means that what you are responsible for in this section is the first two sentences of Theorem 5 and everything from Example 4 to the end of the section. For exam-problems related to this section, I would only ask you to deal with power series centered at 0 (i.e. with x0 = 0). Anything covered in homework, both book- and non-book problems. Anything I said in class, whether or not it was covered in the book or in homework.
Review session	I will hold a question-and-answer review session (6th- and 7th-period sections combined) Sunday starting at 2:30 p.m. in the location I announced in class (if you were not there, ask a classmate for directions).
Viewing and/or picking up graded exam	The final exams are now graded. Please remember that *I will not communicate any* grade-related information by email**; email that asks grade-related questions will not receive a response. Students wishing to know their final-exam scores and/or wanting to see their graded finals should see me in my office in January. Once I know my January office hours, I will post them on my schedule page.

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