• Read the Class home page and Syllabus and course information.
• Go to the miscellaneous handouts page and read the web handouts “Taking notes in a college math class” and “What is a solution?”.
• Read the first paragraph of “To the Student” on p. xi.
• Read Section 1.1.
• 1.1/ 1-8, 11-13

• Read Section 1.2.
• 1.2/ 1-6, 17-20, 25-28, 30, 50
• Read Section 2.1.1

• Read Section 2.1.2
• 1.1/ 9-10
• Read what’s currently on the (just-created) page “Some mistakes in Zill & Wright”. I will add to this page as we go along, if I find additional mistakes.
• 1.2/ 31ac, 32

• For the DE we discussed at length in class on 1/20/16,
  $$ \frac{dx}{dt} = x^2-4,$$
  one of the ways that we wrote the set of all solutions was
  $$\left\{x(t)=\frac{2(1+Ce^{4t})}{1-Ce^{4t}}\right\}\ \ {\rm and}\ \ x(t)=-2.$$

  Using this, find the solution of the initial-value problem for this DE with each of the following initial conditions: (a) \( x(0)=1\); (b) \(x(0)=-2\); (c) \(x(0)=-3.\) In each case, state the domain of the solution.

• Show that the set of all solutions of the DE

  $$ \frac{dy}{dx} = y(1-y^2)$$

  can be written in each of the following forms:

  $${\rm (1)}\ \ \left\{ \ln\left(y^2\left|\frac{1-y}{1+y}\right|\right) = 2x + C \right\}\ \ {\rm and}\ \ y \equiv 0 \ \ {\rm and}\ \ y \equiv 1 \ \ {\rm and}\ \ y \equiv -1,$$

  $${\rm (2)}\ \ \left\{ y^2(1-y) =Ce^{2x}(1+y)\right\}\ \ {\rm and}\ \ y \equiv -1,$$
  and
  $${\rm (3)}\ \ \left\{ e^{2x}(1+y) =Cy^2(1-y)\right\}\ \ {\rm and}\ \ y \equiv 0 \ \ {\rm and}\ \ y \equiv 1.$$

  In each of (1), (2), and (3), the collection of functions in curly braces is a 1-parameter family of implicit solutions. The equations within these braces cannot be solved explicitly for \(y\) in terms of \(x\). Above, the equations \( y \equiv 0, y \equiv 1,\) and \( y \equiv -1\), mean the constant functions defined by \( y(x)=0\ {\rm for\ all}\ x\), \( y(x)=1 \ {\rm for\ all}\ x\), and \( y(x)=-1 \ {\rm for\ all}\ x\), respectively; the symbol “≡” in these equations is read “is identically equal to.”
      The WordPress plugin I’m using to typeset mathematical formulas or sentences is making everything come out in boldface. The boldfacing of the “and”s and “for all”s above is not for the sake of emphasis.

• 2.2/ 4-17, 23-27, 29, 49, 50. Read the following first:
  • Change the instructions for the problems 1-22 to: “Find all solutions, including those that are missed by the method of separation of variables.” (For some, but not all, of the differential-form DEs in the problems in Section 2.2, there are solutions that separation of variables misses.) Also, for DEs in derivative form, explicit formulas for solutions are preferable to implicit solutions, provided it is algebraically possible to find explicit formulas. (For example, if the dependent variable is y and independent variable x, an equation of the form y=φ(x) is preferable to an equation of the form G(x,y)=0 or H(y)=G(x), assuming that it is possible to solve these equations for y in terms of x.) In such cases, express your answers in the form “dependent variable = explicit function of independent variable.” The book’s answer to 2.2/7, for example, is one that I would not give full credit for. In this problem, in my class you are not finished until you solve for y in terms of x.
  • With the instructions modified as above, the book’s answer to #17 is incomplete. I have skimmed, but not carefully checked, the book’s answers to the assigned problems; I may have missed some besides #7 and #17 whose book-answers I would not be satisfied with. I haven’t looked at the Student Resource Manual at all, so I can’t express an opinion what’s in there. My past experience has been that solutions manuals are unreliable, unnecessary, and have the potential to mislead students. Their main effect (and purpose) is to enrich publishers.
  • Problem #27 is an example of a “hybrid IVP”, a format that I consider to be poor. Choosing to write a DE in differential form from the start, not as a step in attempting to solve a DE that started in derivative form, means that you are starting with a problem in which neither variable is preferred to the other as the independent variable. The correct analog of “initial condition” for “DE in differential form with variables \( (x,y) \) is a point \((x_0,y_0)\) through which the solution-curve is required to pass. Writing the initial condition in the form \(y(x_0)=y_0\) indicates that the writer does have a preference for which variable is which (\(x\) independent, \(y\) dependent). That being the case, the writer should have written the original DE in derivative form, not differential form.

• In problems 1-24:
  • You may ignore the instruction pertaining to transient terms.
  • Assuming you make no mistakes in setting up the integrals that need to be done, all of the integrals needed for these exercises are doable by techniques covered in Calculus 1 and 2, but the integrals are not all the easiest of integrals. These exercises have a twofold purpose: to get you used to the method of integrating factors, and to refresh your memory of integration-techniques.
  • The second sentence of the instructions is analogous to saying “Tell me what class you’re taking” to someone who may be taking more than one class. For several of the DEs in problems 1-24, there is more than one largest interval over which a general solution is defined. (“Largest” here means “most inclusive”; an interval \(I\) is larger than an interval \(J\) if \(I\) wholly contains \(J\). “Larger”, in this context, has nothing to do with the length of an interval. The intervals \((0,\infty)\) and \((1,\infty)\) both have infinite length, but the first is larger than the second, in the sense being used here. In that same sense, given two intervals \(I\) and \(J\), neither of which is a subset of the other, we do not say that either is larger than the other.)

        Therefore, change the instruction “Give the largest interval \(I\) over which the general solution is defined” to “Give the largest interval(s) \(I\) over which general solution(s) is/are defined.”

  • Since the book’s “largest interval” instruction does not make sense as written, for several of the exercises it is not clear what question the authors intended to be answering in the back of the book. With the instructions for 1-24 changed as above, several answers in the back of the books are wrong. For example, in #7, the book gives only the general solution on one of the largest intervals for this DE, the interval \((0,\infty)\). There is another largest interval on which there is a general solution in the sense defined on p. 57 and p. 10, the interval \((-\infty,0)\). The general solution on the latter interval is \( y(x)= x^{-1}\ln|x| + Cx^{-1}\). If the book’s instruction had been “Find a largest interval over which there is a general solution in the sense defined on p. 57 and p. 10,” then the answer in the back of the book (for this problem) would be fine.
  • In doing these exercises, use the meaning of “general solution” that I’ve been using in class: the set of all solutions (where the domain of each solution is taken to be as large as possible; if there is more than one largest interval with a general solution, find them all). There are several problem’s with the book’s definition of “general solution”; see the “Some mistakes in Zill & Wright” page.
  • There is also a problem with the book’s definitions of “general solution” on p. 57 and p. 10. On p. 57, the book defines “general solution” only for first-order linear DEs that are in standard form (equation (2) on p. 54, not equation (1)). Several of the DEs in these exercises are not in standard form, so the definition on p. 57 doesn’t apply. The authors do not make clear whether, by “general solution” of a DE of the form in equation (1), they mean “general solution of the DE you get after dividing through by \( a_1(x)\).” This appears to be what the authors mean, but there may be solutions (and whole families of solutions) on intervals that are larger than the largest intervals on which the functions \( P\) and \( f\) in equation (2) that you’d get from dividing equation (1) by \( a_1(x)\) are defined.
  • In #15, first you have to rewrite the equation in derivative form, figuring out which choice of dependent/independent variables makes the resulting equation linear. Once you do this, there are two largest intervals on which the set of all solutions is a 1-parameter family. The book’s answer gives only the interval \( (0,\infty);\); the other is \( (-\infty,0) \). However, for this DE, there is a perfectly good set of solutions on every open interval, and the largest such interval is the whole real line, \( (-\infty,\infty)\). The set of solutions of this DE on \( (-\infty,\infty)\) (or on any open interval containing 0) is
    the two-parameter family

    $$\left\{ x(y)= \left\{ \begin{array}{ll} 2y^6+c_1 y^4 &\mbox{if}\ y\geq 0, \\ 2y^6+c_2 y^4 &\mbox{if}\ y< 0,\end{array}\right. \ \ c_1, c_2\ \mbox{arbitrary constants}\right\}.$$

    It is not clear whether Zill and Wright would call this "the general solution on \( (-\infty,\infty)\)", or would say that this DE does not have a general solution on \( (-\infty,\infty)\). (The definition on p. 10 of “general solution on an interval \(I\)”, applied to a first-order DE, disallows the term “general solution on \(I\)” if the set of all solutions is a family with more than one parameter.)

  • Solutions to linear equations should be expressed in explicit form: (dependent variable) = (formula in terms of independent variable). For this set of problems, several answers in the back of the book are not expressed this way.
  • In #21, you should find that there are infinitely many largest intervals on which the set of solutions is the 1-parameter family given by the formula in the back of the book (but with \(r\) solved for explicitly in terms of \(\theta\)).

A given DE is not exact unless (1) it is in differential form, and (2) it has an exact differential on one side of the equation, and zero on the other. The book’s Definition 2.4.1 is correct (except that on the next-to-last line, it should say “exact equation on R“, and the last sentence should have the words “in R” or “on R” inserted at the end). The only other equations that are exact are those of the form “0= (exact differential)” instead of “(exact differential) = 0”. It is good for students to get practice with non-exact DEs that can “easily be turned into” exact ones, and then solved. (Here I’m intentionally being vague about what “easily” and “turning one equation into another” mean; a precise statement would require a long digression that you would probably not find illuminating.) Several of the DEs in 1-20 are of this type, and so is the DE in Example 3, p. 67 (but there’s a trick in this example that’s not represented in exercises 1-20). Not every first-order DE you’ll be expected to solve will be separable, linear, or exact. However, do not be misled by the discussion in Example 3. The original DE is not exact, and “clearing denominators and putting everything on one side of the equation” is a technique that is sometimes helpful and sometimes not. This technique should be in your arsenal of tools as something that can potentially be helpful, but you shouldn’t think of DEs for which it helps as a special class of DEs (like separable DEs, linear DEs, and exact DEs), and you should not confuse them terminologically with exact DEs.

• Read the handout “A terrible method for solving exact equations”. The parenthetic “we proved it!” doesn’t apply this semester, but everything else does.
• Work through the old exam I handed out. Try to solve problems completely. Don’t stop at the “Yeah, I know how to finish up from here” mark. Stopping at that point will give you an unrealistic assessment of how long it takes you to do problems and how well-prepared you are. It may cause you to overlook subtleties in one or more problems; devils may be hiding in the details.

     Also remember that when asked to solve a DE in derivative form, say with dependent variable \(y\) and independent variable \(x\), and especially for an IVP for such a DE, I expect your final answer to give an explicit (not implicit) formula for \(y(x)\) whenever possible. It will not always be possible. In your homework, one of the things you should be practicing is telling the difference between which equations \(G(x,y)=0\) can be solved algebraically for \(y(x)\), and which cannot.

Note: We did not cover the last part of Section 2.4, which starts near the bottom of p. 67: “integrating factors” in the context of DEs in differential form. This notion of integrating factor is different from the notion of integrating factors for linear DEs (but can be viewed as a generalization of the linear-DE notion). You are not responsible for this part of Section 2.4 (at least not on this exam). So, for example, you would not be expected to be able to solve the DE in Example 4 on p. 68.

• Read Section 4.3 (we are skipping Section 4.2 for now). In “Case II” on p. 133, you may ignore the justification given for why equation (6) is the general solution. (The given justification is in Section 4.2, but we will justify this result in class a different way.)

  Note: there are two common terminologies for equation (3) on p. 133. One is the auxiliary equation, which is the term used in this textbook. The other is the characteristic equation.

• Go through your returned exam:
  • Make sure you understand all my comments.
  • Redo every problem you did not get completely correct.
  • As I’ve mentioned, DEs and IVPs come with built-in checks on your answers. Check whether all your solutions to DEs or IVPs on the exam are, in fact, solutions to the given DE or IVP. This is something you should always be doing in your homework problems. Note that you do not need to be given answers in order to do this. (This is an important fact.) In your homework, if the only “checking” you are doing is looking to see whether your answers to odd-numbered problems agree with the ones in the back of the book, you are not getting the full benefit of the exercises. A large number of students would have done far better on the exam had they been in the habit of doing this sort of checking. Note: the checking I’m talking about may not tell you whether you’ve found all the solutions of a DE, but it will tell you whether every function you’ve found is a solution.

• 4.1/ 31-35
• Finish reading Section 4.4.

• 4.4/ 1, 2, 9, 12, 27, 31, 32. See notes below.

  Notes:

  1. To do problems 1, 2, 9, and 27 by the methods covered in class as of Monday 2/22, you have to recognize for a DE of the form \( L[y]={\rm constant}\), the right-hand side can be viewed as \( {\rm constant}\times e^{\alpha x}\) with \(\alpha =0\).
  2. To do problem 32 by the methods covered in class as of Monday 2/22, you have to write the right-hand side as a linear combination of exponential functions, then use the superposition principle for linear equations (Theorem 4.1.7, p. 126; also review Example 11 on that page).
• Unfortunately, quite a few of the exercises in Section 4.4 require the use of superposition and/or use of particular-solution guesses that I am building to gradually. Therefore, a lot of these exercises will not be doable by what I’ve covered in class until I’ve covered the entire section. The book’s treatment is not thorough or precise enough for me to be satisfied that mirroring the book’s presentation would leave you with adequate skills for applying the Method of Undetermined Coefficients. However, I have already had you read this section for homework, so you already may be able to do more exercises than are due Wednesday. If so, I highly recommend that you get started on these, since the last HW assignment for this section is likely to be a “balloon assignment” with a very large number of exercises that I could not previously assign based on what had been covered in class up through previous lectures.

Notes:

1. These problems are doable using the various lines of the chart I put on the blackboard on Wed. 2/24/16, plus (in some cases) superposition. I have not done any MUC examples in class yet where superposition is necessary. I also have not yet done any examples in which \(g(x)\) is a polynomial with more than one term, or in which 0 is a root of the characteristic polynomial, but what I wrote in the chart covers those cases.
2. In the MUC, if the right-hand side of \(L[y]=g\) is a sum \(g_1+g_2+\dots +g_k\), group together all the \(g_j\) with the same “\( \alpha + \beta i \)” before applying superposition. Otherwise you will unnecessarily increase the amount of work you need to do.
3. In #s 33 & 34, assume that \(\omega\neq 0\).
4. In #34, do not overlook that there are two cases that must be dealt with separately: \(\gamma\neq\omega\) and \(\gamma=\omega\).
5. Note that you do have an assignment due Mon. 3/7/16, the day after spring break. I would have preferred being able to include that assignment as part of the assignment due 2/26/16, to give you a freer spring break, but we have not gotten far enough in class for that. (We’re not going slowly; the MUC always takes me about three lectures to cover.) If you feel able to do these other problems based just on your reading, then you’re certainly welcome to get them done before spring break. The book’s presentation is different from mine, so if you do these problems before I present that material in class, you will be thinking about them differently from the way I’ll be presenting them–but you still should be able to do them based on the book’s presentation (if the inaccuracies mentioned on the entry I’ve added to “Some mistakes in Zill & Wright” for pp. 143-146 don’t mess you up). But regardless, you’ll need to have them done by the time you return from spring break, in order to have adequate time to study for the 3/11/16 midterm.

• 4.4/ 21, 25, 26
• Do these additional exercises on the Method of Undetermined Coefficients.

• Re-read pp. 135-136, from the paragraph “Higher-Order Equations” through just before the paragraph “Use of Computers”. The paragraph before “Use of Computers” discusses the Rational Root Theorem (without mentioning that name), a theorem that enables you to find at least one root of a polynomial of arbitrary degree when that polynomial is “cooperative” enough to have a rational root. For cubic polynomials with at least one rational root, this enables you to find a complete factorization (by methods such as the one in this paragraph of the book). The characteristic/auxiliary polynomials \(p_L(r)\)for the higher-order DEs in the exercises below are all cooperative enough either in this sense, or in one of the others discussed in class, for you to completely factor the polynomials.
• 4.3/ 16, 17, 20–26, 35, 36.
• 4.4/ 22–24, 35, 36.
• 4.5/ 27, 33, 34, with the instructions changed to the following: for each differential operator \(L,\) find a fundamental set of solutions of the equation \(L[y]=0.\)

Remark (ii) at the end of the section says, “[D]o not hesitate to simplify the form of \(y_p\).” This should be said more emphatically and more accurately: Always simplify your formula for the general solution as much as possible. See the comment concerning p. 161 in “Some mistakes in Zill & Wright”.

• 4.6/ 1-6, 10, 12, 15, 17, 23. Note: for any of these DEs in standard form (\({y’}'(x)+p(x)y'(x) +q(x)y(x)=g(x)\)), we consider only domain-intervals on which \(p(x), q(x),\) and \(g(x)\) are defined at every point. The method of Variation of Parameters yields all the solutions on an interval on which the functions \(p, q,\) and \(g\) are continuous. In general this is a more restrictive condition than just being defined at eery point, but the functions you’ll see in these problems are continuous on every interval on which they’re defined. The answers in the back of the book for this section generally omit mention of the domain intervals. The answer given for #1, for example, is correct on any interval of the form \(( (n-\frac{1}{2})\pi, (n+\frac{1}{2})\pi)\), where \(n\) is an integer. The set of all maximal-domain solutions is the collection of all such two-parameter families of solutions (one family for each \(n\)).
• Of the problems in Section 4.6 assigned above, three of the problems on the list above can be done using the Method of Undetermined Coefficients (MUC). Find which ones these are, do them with the MUC, compare your answers with the answers you found by Variation of Parameters (they should be equivalent), and decide which method you think was easier for these problems.

Note: On exams, I rarely say, “Solve this DE by this specific method” (although I do say this occasionally when I am testing on one method we haven’t gotten to yet). Almost always, I expect students to be able to recognize which method(s) could be used to solve a DE I’m asking them to solve, and not to solve a DE by an unnecessarily complicated method when a simpler one would work. (See the second paragraph of instructions on the recent midterm.)

• 4.Review (pp. 190-191)/ 38
• Usually when we’ve just covered a new method for solving DEs, I assign some exercises in which you have to find general solutions, and some in which you have to solve an initial-value problem. The initial-value problems in the exercises for Section 4.4 are 19–22. Figure out why I am not assigning these, but am assigning 4.Review/ 38.

1. For now, solve 1–14 and 31–32 only on the interval \((0,\infty) = \{x>0\}\). For these problems, any answer in the back of the book that includes the expression “\(\ln x\)” is wrong on the interval \((-\infty,0)=\{x<0\}\). In the next assignment, you will modify your answers to get the general solutions on both the interval \((0,\infty)\) and the interval \((-\infty,0)\).
2. The method by which the book says to do exercises 31–36 is the method presented in class on 3/21/16. It works perfectly well for exercises 1–14 and for all Cauchy-Euler equations. There are two ways of approaching Cauchy-Euler equations, both of which rely on what I called the “indicial equation” (“auxiliary equation” in the book). There is a quicker way to derive the indicial equation than by the change-of-variables I used on 3/21/16, which I didn’t get to on 3/21 but will get to on 3/23. But once you have the indicial equation, both methods of approaching Cauchy-Euler equations give you the same fundamental sets of solutions, so exercises 31–32 could just as well have been grouped with 1–18. The main difference in the two approaches is seen only in non-homogeneous equations.

• Modify your answers to 4.7/ 1–14, 31, 32 so that they give general solutions on both the interval \(\{x>0\}\) and the interval \(\{x<0\}\).
• 4.7/ 19–24, 33–35, 37–38. I suggest ignoring the book’s instructions as to what method to use for these, since I will give not give any such instructions or hints on exams. Rather, choose the method that you think is best (or that you find the simplest) for approaching each problem. To get a feel for which method you think is best, you may want to try doing several of these problems two ways (if they can be done more than one way that you know).
• Optional: see comments added to “Some mistakes in Zill & Wright” concerning p. 163 and p. 168.

• Read Section 7.1. As you get to each of pages 274, 275, 276, 277, and 278, read the comments for these pages in “Some mistakes in Zill & Wright”. (The latter is not optional. This section of the book is very poorly written. Among other things, the notation used there can do actual damage to students, causing or reinforcing misunderstandings that can persist long after this course is over. In class and on exams I will be using good notation, not what’s in the book, and I will expect you to understand the notation I use.)
• 7.1/ 11–14, 27, 29, 30. The notation \({\mathcal L}\{f(t)\}\) in the instructions should be changed to \({\mathcal L}\{f\}\). In problems 27, 29, 30, the only parts of Theorem 7.1.1 (which should also be restated with correct notation) you’ll need to use are the ones we covered in class on Friday 3/25/16, namely parts (a) and (c). However, you’ll also need to use the linearity property of the Laplace Transform discussed on p. 276, but which we did not get to in class on Friday. “Some mistakes in Zill & Wright” has a more careful and precise (but lengthy) discussion of this property.

• 7.1/ 1–10, 31, 32, 37, 38, 52. For 37–38 (in addition to 31–32), complete the problem by using Table 7.1.1.
• 7.1/ 39, 40. Do this two ways: (i) using the method suggested in the book (plus Table 7.1.1); (ii) using two general properties of Laplace transform derived in class on Wed. 3/30/16 (plus Table 7.1.1). The relevant general properties are stated in the book as exercise 7.1/54 and Theorem 7.3.1 (p. 290).
• Read Section 7.2. But as soon as you read the portion of this section that’s on p. 281, read the corrected definition of inverse Laplace transform in “Some mistakes in Zill & Wright” (in the comment regarding p. 281).

• 7.2/ 35–38, 41, 42
• 7.3/ 1, 4, 7–11, 13, 15–17, 19, 21, 23, 24, 26, 30. See note below.
• 7.4/ 1, 2, 7, 11. (We covered Theorem 7.4.1 in class several lectures ago.)

Note: In all problems in sections 7.3 and 7.4 in which you are asked to solve an initial-value problem, it is fine to use a table of Laplace transforms. In fact, using a table for these problems is preferable, so that the time spent on these problems is focused on the method:

1. Transforming a differential equation for one function \(y\) into an algebraic equation for the Laplace transform \(Y={\cal L}\{y\}\);
2. solving this algebraic equation to get \(Y(s)=\) (specific formula);
3. rewriting that specific formula as a linear combination of expressions in the “\(F(s)\)” column of a Laplace-transform table (restricting to the rows for which we’ve actually derived the indicated formula); and then
4. inverting the transform to get a formula for \(y(t)\).

The inside back cover of your textbook has a Laplace-transform table with a lot more formulas than you will need. If you’ve used the work done in class to construct your own table (combining the mini-tables I’ve put on the board in several lectures), you’ll have all the formulas you need. On your remaining exams, you will be given a Laplace-transform table that’s shorter than the one in the back of the book, with fewer (but still some) extraneous formulas.

Warning: In Laplace-transform tables, if you see the symbol “\(*\)” (as in \(f*g\)), it does not mean multiplication. It means convolution (see Section 7.4.2), an operation we have not yet discussed (and may not get to).

• 7.3/ 66–70. In these problems and the ones from Section 7.4 below, put your final answers in “tabular form”.

  Note: On my exams, I rarely give DEs with the right-hand side already expressed for you in terms of unit step-functions, as they are in problems 67, 68, and 70. Rather, I give the right-hand sides of these equations in “tabular form”, expect you to be able to convert these to expressions involving unit step-functions (if Laplace transform is the method you’ll be using), and then convert your final answers back into tabular form. The whole point of the Laplace-transform method for solving IVP’s is to be able to solve problems of this type from start to finish. When Mother Nature presents you with a system you have to model by setting up a differential equation, she doesn’t say, “And here’s how to express my forces/voltages/whatevers in terms of unit step-functions, dearie.”

• 7.4/ 13, 14.
• Re-read Section 6.1.

• 6.1/ 11–15, 19–22, 23–26, 29. Change the instructions for 23–24 and 25–30 to: “… so that the summation index is the power of x in each term.” It does not matter what letter is used for this index (as long as it’s not a letter already being used for something else in the same formula).
• Start reading Section 6.2. Once you get as far as the top of p. 240, read the comment on p. 240 in “Some mistakes in Zill & Wright”.

• Finish reading Section 6.2.
• 6.2/ 3, 4, 7, 9, 11, 12. In all of these, you should be able to figure out formulas (or patterns) for all the coefficients, as in Examples 5 and 6 in the book. In this regard, the answer in the back of the book for #9 is not satisfactory; it does not make clear what the pattern is for the numerators of the coefficients.

  In general, the notation “\(+ \dots\)” should be used only when the pattern of the omitted terms has been made clear by the pattern of the explicitly written terms. There is other notation that is preferable when the writer does not know what the pattern is. In Example 7 in the book, for instance, in the formulas for \(y_1(x)\) and \(y_2(x)\), instead of “\(+ \dots\)” it would have been better to write “\(+\ O(x^6)\)”, which in the power-series context means “+ terms of order 6 and higher”.