# Homework Assignments MAP 2302, Section 3146—Elementary Differential Equations Spring 2010

Last update made by D. Groisser Sat Aug 28 00:47:17 EDT 2010

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, Fundamentals of Differential Equations and Boundary Value Prolems, 5th 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 just read 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
F 1/8/10
• Read the syllabus and the web handouts "Taking notes in a college math class" and "What is a solution?".

• Read Section 1.1 and do problems 1.1/ 1-16.

• Do non-book problem #1.
• M 1/11/10 1.2/ 1, 3-5, 14, 15, 19-22. "Explicit solution" is synonymous with "solution". I will say more about the terminology "explicit solution" and "implicit solution" (which we have not used yet in class) at a later time.
Note: Many of your homework assignments will be a lot longer than the ones I've given so far. I don't want anyone to feel after Drop/Add that he/she wasn't warned. Often, most of the book problems in a section aren't doable until we've finished covering practically the entire section, at which time I may give you a large batch to do all at once. Heed the suggestion above the assignment-chart: "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."
W 1/13/10
• Read the material on "implicit solutions" in Section 1.2, pp. 8-9.
• 1.2/ 2, 10, 11, 30
• Do non-book problem #2.
• Read sections 2.1 and 2.2. (we are skipping sections 1.3 and 1.4). In your reading, make sure you understand the discussion following Example 3 (bottom of p. 44 to top of p. 45).
• F 1/15/10
• 1.2/ 18,23-29
• 2.2/ 1-5,8,9,11,17-19. "Solve the equation (or IVP)" means "Find the general solution of the equation (or IVP)", i.e. the collection of all solutions to the DE (or IVP). For IVP's, the "general solution" is usually a single function (this the case whenever the hypotheses of the Fundamental Existence/Uniqueness Theorem—Theorem 1 on p. 12—are met); when no initial conditions are imposed, the general solution is usually an infinite collection of functions.
• W 1/20/10
• 2.2/13,23, 27abc, 28-31, 33, 34
• Exercises 3-6 on the updated non-book problems page. (These have changed since the last time you looked at this page.)

In doing the exercises from Section 2.2 or the non-book problems above, you may find that the hardest part is doing the integrals. I'm intentionally assigning problems that require you to refresh your basic integration techniques. Remember my warning from the first day of class (which is also on the class home page): You will need a good working knowledge of Calculus 1 and 2. In particular, you will be expected to know integration techniques ... .If you are weak in any of these areas, or it's been a while since you took calculus, you will need to spend extra time reviewing or relearning that material. Mistakes in prerequisite material will be graded harshly on exams.

Your integration skills need to be sufficient for you to get the right answers to problems such as the ones in this homework assignment, not merely to go through the motions. One type of mistake I penalize heavily is mis-remembering the derivatives of common functions. For example, expect to lose A LOT of credit on an exam problem if you write ``∫ lnx dx=1/x'', or ``(d/dx)(1/x)= lnx'', even if the rest of your work is correct. (1/x is the derivative of lnx, not an antiderivative; lnx is an antiderivative of 1/x, not its derivative.)

This does not mean you should study integration techniques to the exclusion of material you otherwise would have studied to do your homework or prepare for exams. You need both to review the old (if it's not fresh in your mind) and learn the new, even if this takes a lot of time.

• F 1/22/10
• Read Section 2.3. When you apply the method that's in the box on p. 51, don't forget the first step (writing the equation in "standard form"). Be especially careful to identify the function P correctly; its sign is very important. For example, in 2.3/17, P(x)=-1/x, not just 1/x.

• 2.3/ 1-6, 7-9, 13-15, 17-20, 33. 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 (that's why I warned you above to be careful when identifying the function P. 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.
• M 1/25/10
• On this day, remind me to discuss whether to keep the first midterm on its originally scheduled day (Mon. Feb. 1), or move it to a later day.
• 2.3/ 22,23 (re-read Example 2 on p. 52 first), 24, 27a, 28, 30-32, 35.
• Do non-book problem #7.
• As I mentioned in class several lectures ago, I am not satisfied with the book's treatment of "implicit solutions" and equations in differential form. The good news is that I'm writing my own notes on these topics for you. The bad news is that the notes are already longer than I expected, and I haven't even gotten to the part where I will define "implicit solution". However, I'm assigning you now to read the part that I've written so far, so that when the rest is written you won't have to read everything all at once. I will want you to have read the completed set of notes before your first exam (which, of course, means that I will have to finish writing the notes pretty soon). The notes are here.
• (Continuation of 2.3/ 33b). Consider the differential equation
(*)       xy'-2y=3x
(the same DE as in 2.3/ 33b).
• Without finding any formulas for solutions of (*), show that every solution of (*) defined on a domain that includes 0 satisfies y(0)=0.
• Let C1 and C2 be constants, and define a function φ on the whole real line by
φ(x) = C1x2 -3x for x≥0,
φ(x) = C2x2 -3x for x<0.
1. Show that for every choice of the constants C1 and C2, the function φ above is a solution of (*) on the whole real line.
2. Show that the general solution of (*) on the interval (0,∞) is given by the formula
y=Cx2-3x, where C is an arbitrary constant.
3. Show that the general solution of (*) on the interval (-∞,0) is given by the same formula, y=Cx2 -3x. Why are these solutions not the same functions as in the preceding part of this problem, even though the formulas are the same?
4. Show that the collection of functions φ defined earlier in this problem, treating C1 and C2 as arbitrary constants (i.e. allowing all real numbers for their values), is the general solution of (*) on the whole real line.
Note: in this problem, the domain-interval ``the whole real line'' may be replaced by any open interval containing 0; the facts in parts 1 and 4 are true for any such interval.
• W 1/27/10
• Read Section 2.4 through the "Test for Exactness" on p. 61.
• 2.4/ 1-8
• [This part is not due till Friday since I am not posting it until late Tuesday night, but if you see it early enough you can get a headstart.] Read the updates to the notes posted in the previous assignment. The updates are as follows:
• A footnote has been added to the last part of Definition 2.10 (p. 10).
• Definition 2.11 (p.11) has been expanded. This, and the remaining changes, affect the page-breaks of the last few pages of the first version of these notes. (So the corresponding pages may look different, but the only new material is what I'm telling you here.)
• Definition 2.12 (p.11) has been altered (the first part now defines solution curve, not solution) and expanded.
• On p. 12, the paragraphs immediately before and after Remark 2.13 are new, and a typo in Remark 2.13 has been corrected.
• Everything from ``Now we return to Definition 2.12 ...'' on p. 14 to the end of the notes is new.
• F 1/29/10
• Read the updates to the notes. For updates through Tues. (1/26), see the list given with the homework due 1/27. If I make any other updates before Friday, I will similarly list them here, so that you don't have to re-read the whole set of notes.

• Here is information I should have given regarding 2.4/ 1-8 in the previous assignment:
Classify an equation in differential form as linear if at least one of the related derivative-form equations (the ones you get by formally dividing through by dx and dy) are linear. It is possible for one of these derivative-form equations to be linear while the other is nonlinear. This happens in exercises 1,2,5,6 and 7. For example, #5 is linear as an equation for x(y), but not as an equation for y(x).
• 2.2/6, 16. An equation in differential form is called separable if, by the operations of addition/subtraction of differentials, and multiplication by functions (other than the constant function 0), you can arrive at an equation of the form g(x)dx=h(y)dy (assuming the variables are x and y). This is equivalent to the condition that the derivative-form equation obtained by formally dividing the original equation by dx or dy is separable.

• 2.2/ 22. Note that although the differential equation doesn't specify independent and dependent variables, the initial condition does. Thus your goal in #22 is to produce an explicit solution "y(x)= ...". But this exercise is an example of what I call a "schizophrenic" IVP. In practice, if you are interested in solutions with independent variable x and dependent variable y (which is what an initial condition of the form ``y(x0)=y0'' indicates), then the differential equation you're interested in at the start is one in derivative form (which in exercise 22 would be x2 +2y dy/dx=0, or an algebraically equivalent version), not one in differential form. Putting the DE into differential form may be a useful intermediate step for solving such a problem, but differential form is not the natural starting point. On the other hand, if what you are interested in from the start is a solution to a differential-form DE, then you should not express a preference for one variable over the other by asking for a solution that satisfies a condition of the form ``y(x0)=y0'' or ``x(y0)=x0''. Instead, you should ask for a solution whose graph passes through the point (x0, y0), which in exercise 22 would be the point (0,2).

• Read the portion of Section 2.4 from ``Method for Solving Exact Equations'' (p. 63) through the end of the section.

• 2.4/11, 12, 14, 16, 17, 19, 20-22, 27a, 28a, 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). Note that exercises 21-26 are "schizophrenic IVPs". In all of these, the goal would be to find an explicit formula for a solution—if algebraically possible—with the choice of independent/dependent variables indicated by the initial condition. However, if the equation ``F(variable1, variable2)=0'' that you get via the exact-equation method (in these exercises) is impossible to solve for the dependent variable in terms of the independent variable, you have to settle for the implicit solution.
• M 2/1/10
• Read everything in Section 2.4 that you have not yet read.

• 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. (In your reading of Section 2.4, you've seen at least part of proof, but the book leaves the key part of the argument as an exercise that relies on a theorem that you are unlikely ever to have seen.)

• Do non-book problems 8 and 9. These two problems have been added to the non-book-problems page since the last time you looked at it. Note: problem 9 has parts (a) through (j).

• Read the updates to my notes. Here are the changes since the last update:
• The definition of the term ``algebraically equivalent'' (which is in the same location on p. 8 as before) has been put inside a boldfaced, numbered definition so that it could be more easily found and referred to later. This affects the numbering of later items (old Definition 2.8 = new Definition 2.9) and the page-breaks beyond this point.
• The paragraph after Remark 2.14 (old Remark 2.13) has been lengthened.
• A footnote has been added on (new) p. 15. This affects the numbering of later footnotes.
• On p. 16, an existence/uniqueness theorem for DEs in differential form has been added, along with a paragraph before and after it.
• The title of Subsection 2.2.2 has had words added for clarification (p. 17).
• Everything from the bottom of p. 21 (starting with the paragraph above Definition 2.21) up to the beginning of Section 2.3 on p. 23 is new.
• The last paragraph of the notes is new.
• W 2/3/10
• So that you may celebrate Groundhog's Day with friends and family, there are no new exercises. But check back later on Tuesday to see if there are updates to my notes for you to read.

• For students who want a supply of exercises to practice with: 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, and the last part of 41.

• General info One student asked how students in my MAP 2302 classes have performed on the first exam in the past. I don't remember most statistics of that sort of the top of my head, but for anyone who's interested, here's how to find it:
• Under ``Course materials'', click on the ``Past classes'' link.
• On the Past Classes page, click on the link to any of my previous MAP 2302 classes. This will take you to the home page for that class, which has links to grade-scale pages. Each grade-scale page has some statistics on each exam, and a link to the list of scores.
• F 2/5/10 First midterm exam (assignment is to study for it)
M 2/8/10 Read section 4.1. (We're skipping Sections 2.5 and 2.6, and all of Chapter 3.)
W 2/10/10 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".
General info The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list of scores so that you may see the grade distribution. Exams will be returned Friday 2/12/10.

Here are some comments on the exam.

F 2/12/10
• Read Section 4.2 through Theorem 2 on p. 172.
• 4.2/ 1,3,4,6-9,11,12,13-17,21-25.
• Add as part(c) to #21: Now solve the same DE by the "integrating factor" method we learned in Chapter 2. Do you expect to get the same general solution that you got using the method in parts (a) (b)? Do you get the same general solution both ways?
• M 2/15/10
• Read the remainder of Section 4.2.
• 4.2/ 2,5,10,18-20, 27-32. Note: The answer to #27 in the back of the book is wrong.

• Read my comments on the exam (this is the same as the link in the "General info" entry above the assignment due 2/12/10).

• Re-do all the exam problems on which you did not get a perfect score. For all but #7, you should be able to check by yourself whether your new answers are correct (some of the answers that would be harder for you to check by yourself are in my posted comments on the exam), but you are welcome to check your answers against my solutions during office hours. For #7, the final answer is 32/9 gm/L. As a matter of policy, I do not post solutions on the internet.

• Solve the initial-value problem for the DE in exam problem #4, but with the following initial conditions: (c) y(0)=3π/4, and (d) y(0)=3π/2.
• W 2/17/10
• 4.2/ 26,33,35,36,37,43,46ab. Note: The answer to #35a in the back of the book is wrong.
• Do non-book problems 10 and 11. The operator L1+L2 is defined by (L1+L2)[f] =L1[f]+L2[f].
• F 2/19/10 4.3/ 1,2,4,6,7,9-12,17,18,21-23,28
M 2/22/10 4.3/ 3,5,8,13-16,24-26,32,33 (students in electrical engineering may do #34 instead of #33),36
W 2/24/10
• Read Section 4.7 through Theorem 6.
• 4.7/ 1-4,5-8,10-13,15,17,19,20,23,27-29.
In 1-8, in addition to doing what the book says, use Theorem 5 to write down the largest interval on which you are guaranteed, without having to solve the IVP, that a solution exists. An important feature of linear equations is that you can predict the domain of the solution (more precisely, predict that the domain of the solution is at least a certain interval) from the coefficient functions and the initial value of the independent variable alone, without having to solve the equation or use the initial values of the dependent variable and its derivative(s). For nonlinear equations, often the domain depends on the initial value of the depenedent variable and its derivatives, and the domain is usually hard to determine without solving the IVP.
Sometimes a linear equation, especially one that is not in standard form (the form in Theorem 5, in which the top-derivative term has coefficient 1) has one or more solutions with a domain that is larger than what Theorem 5 guarantees. The solution for Exercise 19 is an example.
• F 2/26/10
• 4.4/ 9 (note that -9 = -9et), 10,11,14,17
• Add parts (b) and (c) to 4.4/ 9-11,14,17 as follows:
• (b) Find the general solution of the DE in each problem.
• (c) Find the solution of the initial-value problem for the DE in each problem, with the following initial conditions:
• In 9,10, and 14: y(0)=0=y'(0).
• In 11 and 17: y(0)=1, y'(0)=2.
• 4.5/ 3-8,25,26
• 4.7/ 24ab
• M 3/1/10
• 4.4/ 13,18
• 4.5/ 1,2,20,27,28,29,41
• non-book problem 14
• W 3/3/10
• 4.5/ 42. Part (b) of this problem, if done correctly, shows that the particular solution of the DE in part (a) produced by the Method of Undetermined Coefficients actually has physical significance.

• 4.5/ 45. This is a nice problem that requires you to combine several things you've learned.
The strategy is similar to the approach in Exercise 41. Because of the "piecewise-defined" nature of the right-hand side of the DE, there is a sub-problem on each of three intervals: Ileft = (-∞ -L /(2V)], Imid = [-L /(2V), L /(2V)], and Iright = [-L /(2V), ∞). The solution y(t) defined on the whole real line restricts to solutions yleft, ymid, and yright on these intervals.
You are given that yleft(t) is identically zero. Use the terminal values yleft(-L /(2V)), y'left(-L /(2V)), as the initial values ymid(-L /(2V)), y'mid(-L /(2V)). You then have an IVP to solve on Imid. For this, first find a "particular" solution on this interval using the Method of Undetermined Coefficients (MUC). Then, use this to obtain the general solution of the DE on this interval; this will involve constants c1, c2. Using the IC's at t=-L /(2V), you obtain specific values for c1 and c2, and plugging these back into the general solution gives you the solution ymid of the relevant IVP on Imid.
Now compute the terminal values ymid(L /(2V)), y'mid(L /(2V)), and use them as the initial values yright(L /(2V)), y'right(L /(2V)). You then have a new IVP to solve on Iright. The function yright is what you're looking for.
If you do everything correctly (which may involve some trig identities, depending on how you do certain steps), under the simplifying assumptions m = k = 1 and L = π, you will end up with just what the book says: yright(t)=A sin(t), where A=A(V) is a V-dependent constant (i.e. constant as far as t is concerned, but a function of V). In part (b) of the problem you are interested in the function | A(V) |, which you may use a graphing calculator or computer to plot. The graph is very interesting.
Note: When using MUC to find a particular solution on Imid, you have to handle the cases V≠ 1 and V = 1 separately. (If we were not making the simplifying assumptions m = k = 1 and L = π, these two cases would be πV/L ≠ (k/m)1/2 and πV/L = (k/m)1/2.) In the notation introduced last week, using s for the multiplicity of a certain number as a root of the characteristic polynomial, V≠ 1 puts you in the s = 0 case, while V = 1 puts you in the s = 1 case.
The book's answer to part (a) is correct only for V ≠ 1. The value of A for V=1 is π/2. (I'm just telling you this so you can check your answer; you're still supposed to figure out this answer yourself.)
• F 3/5/10 Second midterm exam (assignment is to study for it)
M 3/15/10 Enjoy your spring break!
General info The grade scale for the first midterm is now posted on your grade-scale page, with a link to the list of scores so that you may see the grade distribution.
W 3/17/10
• 4.4/ 1-8, 12,15,16, 19-26
• 4.5/ 9-16
• F 3/19/10
• 4.4/ 27-32
• 4.5/ 17-19, 21,22,23,24,30, 31-36
• M 3/22/10
• 4.6/1, 3-18, 19 (first sentence only)

• 4.7/24cd, 37-44. (In all of these, assume the domain interval is {t > 0}.) Note: to apply Variation of Parameters as presented in class, you must first put the DE in "standard form", with the coefficient of the second-derivative term being 1 (so divide by the coefficient of this term, if the coefficient isn't 1 to begin with).
Note that it is possible to solve all the DEs in 24cd and 37-43 either by the Cauchy-Euler substitution applied to the inhomogeneous DE, or by using Cauchy-Euler just to find a FSS for the associated homogeneous equation, and then using Variation of Parameters for the inhomogeneous DE. Both methods work. I've deliberately assigned exercises that have you solving some of these equations by one method and some by the other, so that you get used to both approaches.

• Redo 4.7/44 by starting with the substitution y(t)=t–1/2u(t) and seeing where that takes you.
• W 3/24/10 Read Sections 6.1 and 6.2.
F 3/26/10
• 6.2/ 1,11,13,14,15-18
• M 3/29/10
• Read sections 7.1 and 7.2.
• 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)
• W 3/31/10
• 7.2/ 13-20 (use table on p. 384; we'll derive these in class Wed.), 21-28, 29a-d,f,g,j, 30
• Read Sections 7.3 and 7.4. When we get to 7.4 in class, I will not have time to review the method of partial fractions; I will assume you know the method when I do examples.
• F 4/2/10 7.3/ 1-10,12,14,19,25,31. In class, we have derived only the first three lines of the Table 7.2 so far, but I want you to start getting practice using all the properties listed in the table.
M 4/5/10
• 7.4/ 1-10,11,13,15,16,20,21-24,26,27,31

• 7.5/ 15,21,22
• CORRECTION! The last fraction I wrote on the board Monday (after class) as part of the answer to problem 6 had the numerator "s". The numerator should have been "s-3".
W 4/7/10 Third midterm exam (assignment is to study for it).
You will be given a Laplace Transform table to use on the exam; you do not have to memorize any transforms or inverse transforms. (But you will be expected to know how to make use of the information in the table.)
General info The grade scale for the third midterm is now posted on your grade-scale page, with a link to the list of scores so that you may see the grade distribution.
F 4/9/10 No new homework.
M 4/12/10
• 7.5/1-8,10,29. To learn some shortcuts, you may want first to read the web handout "Partial fractions and Laplace Transform problems" (pdf file).

• Read Section 7.6 through Example 5.
• W 4/14/10 7.6/ 1,2. If you feel that you are ready to tackle harder problems based on the reading that was due Monday but that I have not yet covered in class (I will on Wednesday), start on the homework due Friday, since that will be a long assignment.
F 4/16/10
• 7.6/3-10,11-18,29-32,33-40. 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.)