# Errata in Farlow's Differential Equations

This web page contains a list of errata (sorted by the page number) in

Stanley J. Farlow: An Introduction to Differential Equations and their Applications, Dover Publications, 2006. (Originally published by McGraw-Hill, 1994.)

If you spot other errors, let me know. I will add them here with due acknowledgment.

Note: This page is not presented in the way of criticism; it's rather the opposite—I like the book quite a bit and I would like to help others to derive the most out of this good and particularly inexpensive textbook.

## Errata

Page 15:
(Spotted by Dr. Bradley) In the text just after equation (2), the $y^{(n)}$ should be $y^{(n-1)}$, as in: where $f$ is a function of $x, y, y', \ldots, y^{(n-1)}$.
Page 23, last paragraph:
(Spotted by Dr. Peercy) The expression $\partial f/\partial y = \frac13 y^{2/3}$ should be $\partial f/\partial y = \frac13 y^{-2/3}$.
Page 26, Exercise 26(b):
(Spotted by Andrew Zuelsdorf) The answer in the back of the book is $\frac{x^3}{3}$. It should be $\frac{x^3}{3}+c$.
Page 36, Exercise #6:
The answer in the back of the book has a typo; the correct answer is $y = ce^{-x^2} + 1/2$.
Page 42, Exercise #20:
(Spotted by Christine Sweigart, reported by Dr. Lo) The answer in the back of the book is garbled. It is meant to be $c(1-y)e^{x^2}=1+y$. Better yet, $y$ may be computed explicitly as $\ds y=\frac{1+ce^{-x^2}}{1-ce^{-x^2}}$.
Page 42, Exercise #23:
(Spotted by Dr. Soane) The solution in the back of the book is correct but incomplete—the constant function $y(x)=1$ is also a solution. The inadvertent omission is brought about by the necessity of dividing by $y-y^2$ to separate the variables; the division is permissible only if $y$ is other than 0 or 1.
Page 43, Exercise #32:
(Spotted by Anthony Santini, reported by Dr. Kogan) The answer in the back of the book is garbled. It should be $y = \Big( 1 - \int_1^x e^{t^2} \,dt \Big)^{-1}.$
Page 43, Exercise #33:
(Spotted by Dr. Stegenga) In the statement of the problem, the $1+y^2$ term is meant to be outside the square root, as in $\frac{dy}{dx} = \sqrt{1 + \sin x} \; (1+y^2).$
Page 43, Exercise #38:
(Spotted by Christine Sweigart, reported by Dr. Lo) The answer in the back of the book is wrong. It should be $y = cx^2 - x$.
Page 45, Exercise #45:
(Spotted by Dr. Mayans) The differential equation to be solved is $y'^2 + y^2 = 1$. The answer $y=\sin(x+c)$ given in the back of the book is correct but incomplete—the constant functions $y(x)=1$ and $y(x)=-1$ are also solutions.

Aside: The graphs of the constant functions $y(x)=1$ and $y(x)=-1$ form the envelopes of the family of functions $y=\sin(x+c)$. According to Exercise 44, the envelope of a family of solutions to a differential equation is also a solution of the differential equation.

Page 45, Exercise #47:
(Spotted by Dr. Mayans) The answer given in the back of the book is off by one hour. The correct answer is 11:23 a.m.

Aside: This is not truly a differential equations problem as it amounts to solving a trivial equation of the form $dy/dt=f(t)$.

Page 50:
(Spotted by Dr. Soane) In equation (5) the rightmost term is missing a negative sign in the exponent. It should be $Q_0 e^{-0.00012378t}$.
Page 50:
(Spotted by Daniel Engbert) The 15,336 in equation (7) should be 15,327.
Page 56, Exercise #17:
(Spotted by Nick Selock with assistance from Samuel Khuvis and Ari Rabe, reported by Dr. Peercy) The problem's description leads to the model $dP/dt = -k$ (with $k = 0.015$). Its placement in this section on exponential decay is confusing. Nevertheless, the answer in the back of the book is correct.
Page 56, Exercise #18:
(Spotted by Momo Mizuguchi) The answer $5e^{0.3\ln2}$ given in the back of the book is correct, however its decimal approximation is 6.15572 which rounds to 6.16, not 6.15.
Page 56, Exercise #20:
(Spotted by Michael Norris) The problem's statement implies that two hours after 10:00 p.m. is 12:00 p.m. This goes against the convention of referring to midnight as 12:00 a.m.

Aside: The Wikipedia article Confusion at noon and midnight presents an informative discussion of this issue.

Page 57, Exercise #24:
(Spotted by Menachem Greenfeld) The answer in the back of the book is missing the word “million”. It should be $5e^{4\ln2}\text{ million} = 80 \text{ million}$.
Page 57, Exercise #27:
(Spotted by Rachana Haliyur, reported by Dr. Peercy) The answer to part (b) should be 2435 A.D.
Page 58, Exercise #34:
(Spotted by Celia Drew and James Loy, and reported by Dr. Dean) The numerical answer in the back of the book is correct but the exponential form of the answer is missing a factor of 24. It was meant to say $24 e^{(0.08)(366)} \doteq \$124,839,000,000,000$. However… James Loy points out that$24 e^{(0.08)(366)} \doteq \$124,839,974,659,582.22$, theorefore the properly rounded answer is $\$124,840,000,000,000$. Page 62, between equations (4) and (5): (Spotted by Celia Drew, reported by Dr. Dean) It says$\mu(x) = e^{3t/50}$. That$\mu(x)$should be$\mu(t)$. Page 64: (Spotted by Jessica Alignay, reported by Dr. Hoffman) In the equation that comes just after equation (10), the$(100 - t^2)$on the left-hand side should be$(100-t)^2$. Page 65, Exercise #1: (Spotted by Momo Mizuguchi) The answer to part (d) in the back of the book is given in “lbs/gal”. The proper expression is “lb/gal”. Page 66, Exercise #4: The answer in the back of the book is garbled. The correct answer is: $\frac{dQ}{dt} + \Big(\frac{5}{100-2t}\Big) Q = 0, \quad Q(0) = 300.$ Page 71, in the middle of the page: (Spotted by Dr. Robert Kouba) We have: “To give meaning to$1/k$, temperature$T(k)$of an object…”. Here the$T(k)$should be$T(t)$. Page 74, Exercise #6: (Spotted by Rachana Haliyur, reported by Dr. Peercy) Recorded temperatures are missing. The answer of 30°F given in the back of the book is not obtainable. The original 1994 hardcover edition—well, one of the hardcover versions, since there are multiple versions of it around—gives the pertinent data. It says: “Suppose the temperature of the can was 45°F after 0.5 hours and 55°F after 1 hour.” With this data, the answer would be 28.33°F. Page 74, Exercise #8: Based on the given data, the decay constant turns out to be$k = (\ln 2)/2$and the time of murder is$t = -2.561912628$. This corresponds to 5:26 pm. In the answer given in the back of the book, the value of$t$is rounded to$-2.6$, consequently the time of the murder is given as 5:24 pm, which is off by 2 minutes. The 5:24 pm value is repeated in Exercise #12 on page 75. Page 74, Exercise #10: (Spotted by Dr. Dovermann) The correct answer depends on several unstated assumptions. Is the cream's temperature the same or different from the room temperature? Does the cream's temperature remain constant as Mary waits, (e.g., is it kept on ice or in a refrigerator,) or is it brought to the table from the refrigerator and allowed to warm? Analyzing the various cases would be an interesting and instructive mini-project. Here is the answer to one of the many scenarios: If Mary's cream remains at room temperature while she waits, then her coffee will be exactly as hot as John's when they begin to drink. Page 75, Exercise #12: The answer in the back of the book is entirely wrong. The correct value for the decay constant is$k=0.2912836670$. The murder occurred at$t=-3.498093089$, that is, approximately at 4:30 pm. Note: The equation that determines$k$has no solution in terms of elementary functions. You need to compute$k$numerically on a calculator. Page 77, Equation (4): (Spotted by Marie Wagner) The malformed fraction$\ds\frac{m^2g}{}$should be$\ds\frac{m^2g}{k^2}$. The hardcover edition has the correct expression; the error was introduced in the paperback edition. Page 80: (Spotted by Dr. Dovermann) A square root sign is missing in equation (12). It should be: $\frac{d\theta}{dt} = \sqrt{ \frac{v_d^2 - v_s^2}{v_s^2} } \; \frac{1}{t}.$ Page 83: Exercise #9: (Spotted by Sophie Peet) The goal of this exercise is to find a function$g(x,y)$so that$dg = \frac{x\,dx + y\,dy}{x^2+y^2}$. The answer$\ln(x^2+y^2)$in the back of the book is missing a factor of 1/2. It should be$g = \frac{1}{2} \ln(x^2+y^2)$. Page 84, Exercise #17: (Spotted by Hailey Lynch, reported by Dr. Dean) The answer in the back of the book is missing a negative sign in the exponent and also suffers from improper rounding. The correct expressions are: $v(t) = \frac{120}{7} \big( 1 - e^{-t\,/\,375} \big) \approx 17.14285714 \big( 1 - e^{-0.002666666667 \, t} \big) \approx 17.1 \big( 1 - e^{-0.0027 \, t} \big).$ In particular, the ship's limiting velocity is approximately 17.1 ft/sec or 11.7 mph. Page 85, Exercise #20: (Spotted by Sean Shields) In the equation given in part (b), the term$\sqrt{2gt}$should be$\sqrt{2g}\,t$, that is, the$t$should not be under the radical. Page 85, Exercise #22: (Spotted by Dr. Dovermann) In part (e) the condition$v(a)=0$should be$v(1)=0$. In part (g) the equation (24f) is not quite correct. It should be: $y(x) = \frac{x}{2}\Big( \frac{x^k}{1+k} - \frac{x^{-k}}{1 - k} \Big) + \frac{k}{1-k^2}.$ Page 86, Exercise #23: (Spotted by Dr. Dovermann) The equation of motion, given in part (b), has an extra minus sign. It should be$\ds \frac{dv}{dt} = - \frac{gr}{R}$. Additionally, the answer for part (b) in the back of the book is off by a factor of two. The correct answer is 42.54 minutes. Page 89, Exercise #28: (Spotted by Sean Shields) In part (b), “drive” should be “derive”. Furthermore, the$h^3$in the numerator should be$h^2$. Page 94, Equation (6): (Spotted by Mayur Darji, reported by Dr. Dean) The$v'$on left-hand side of the equation should be$mv'$. Page 97: (Spotted by Andrew Zuelsdorf) In the line just before equation (17) we have: “If Newton's method with step size h is used to compute N approximations …”. “Newton” in that sentence is supposed to be “Euler”. Page 98, Exercise #20: (Spotted by Menachem Greenfeld) In part (a) the$t=1$should be$x=1$. Page 103, equation (9): (Spotted by Dr. Nathaniel Mays) The equation is missing a$y_n$term on the right-hand side. It should be $y_{n+1} = y_n + \frac{h}{6} \big( k_{n1} + 2k_{n2} + 2k_{n3} + k_{n4} \big).$ Page 116, Exercise #12: (Spotted by Brian Hillsley) The following error exists only in the book's electronic (eBook) edition; the printed version is fine. The statement “…and any linear combintaion$c_1 e^{x} + c_2 e^{2x}$are solutions…” is meant to say$c_1 e^{x} + c_2 e^{-x}$. Page 117, Exercise #30: (Spotted by Dr. Nathaniel Mays) This problem is wrongly listed under the heading of second order equations with missing$y$. It should be grouped with problems 32–37 under the heading of second order equations with missing$x$. Page 117, in the comments preceding Exercise #32: (Spotted by Clayton Lively, reported by Dr. Lo) The substitution$y=v'$should be$y'=v$. Page 117, in the comments preceding Exercise #39: (Spotted by Daniel Engbert) The initial condition$y'(x_0)=y'$is meant to be$y'(x_0)=y_1$. Page 122, Exercise #4: (Spotted by Andrew Zuelsdorf) The answer in the back of the book is$W(f,g) = (m-n) e^{(m+n)x}$. It should be$W(f,g) = (n-m) e^{(m+n)x}$. Page 122, Exercise #9: (Spotted by Faraz Vatan) Each of the problems 1–10 consists of two parts. First, you are asked to determine if the given pair of functions$f$and$g$is linearly independent on the interval$(-1,1)$. Second, you are asked to compute the Wronskian$W(f,g)$. According to a note on page 118, a function pair is linearly independent if one function is not a constant multiple of the other. (You don't really need a Wronskian to determine that.) In Excercise #9, we are given$f(x) = x$,$g(x) = |x|$. Since$|x|$is not equal to a constant times$x$, then this pair is linearly independent, as the answer in the back of the book correctly states. As to the second part, since the function$g(x) = |x|$is not differentiable at$x=0$, the Wonskian$W(f,g)$is undefined at$x=0$. Therefore the answer in the back of the book, which says$W(f,g)=0$, is wrong. Page 122, Exercise #10: This problem asks you to determine if the pair of functions$f(x) = \ln x$and$g(x) = \ln x^2$is linearly indepdenent on the interval$(-1,1)$. The answer in the back of the book is wrong. The correct answer is that the question is meaningless since the natural logarithm$\ln x$is undefined when$x \le 0$. Page 123, Exercise #25: (Spotted by Kaung San, reported by Dr. Peercy) In equation (21) a minus sign is missing in the exponent. It should be: $W(x) = c e^{-\int p(t) \, dt}.$ Page 126, calculations leading to Theorem 3.5: (Spotted by UHM) Each of the integrals$\int p(s)\,ds$in equations (6), (7), (9) should have a minus sign in front. Furthermore, the$r$and$s$variables introduced in the derivation are not truly necessary since we are dealing with indefinite integrals here. The following simpler form of equation (9) should do: $y_2(x) = y_1(x) \int \frac{e^{-\int p(x)\,dx}}{y_1^2(x)} \,dx .$ In any case, as noted at the bottom of page 126, it is easier to carry out the steps that lead to equation (9) than to use that equation directly. Page 128, Exercise #15: (Spotted by Dr. Bradley) The answer in the back of the book has a sign error. It should be $y = -x \cos x + \sin x \ln|\sin x|.$ Page 129, Exercise #18: (Spotted by Justin Korn) In the answer in the back of the book, the$e^{-x^2}$should be$e^{x^2}$. Page 130 forward: The choice of$m$for the name of the variable in the characteristic equation is unfortunate because it conflicts with the symbol for mass in applications to dynamics. This has been rectified later (page 166 onward) by replacing it with$r$. It would have been better to use the symbol$r$instead of$m$right from the start. Page 132, Example #2: In equation (8), the initial condition should be$y'(0)=0$. Page 134, Exercise #23: (Spotted by Dr. Dovermann) There's a misprint in the hint. It should read$x^2 d^2y/dx^2 = d^2y/dt^2 - dy/dt$. Page 137, line 2 from the top: (Spotted by Dr. Sousedik) The term$\frac{ix^3}{3!}$should have a minus sign in front. On the next line it has the correct (minus) sign. Page 138, Example #2: (Spotted by UHM) The characteristic equation is presented as$m^2+1=0$. It should be$m^2+4=0$. Page 139, Exercise #11: (Spotted by Corey Kegerreis, reported by Dr. Peercy) The answer in the back of the book is missing a term. The correct answer is$y = \cos 2x -\frac{1}{2} \sin 2x$. Page 139, Exercise #15: (Spotted by Brian Hillsley) The following error exists only in the book's electronic (eBook) edition; the printed version is fine. The differential equation should be$y'' - 4y' + 7y = 0$. Page 140, Exercise #19: Equation (19) is meant be $xy'' - y' - x^3 y = 0.$ Furthermore (spotted by Menachem Greenfeld) the answer to part (b) in the back of the book needs to be adjusted to$v' = x^2 + \frac{1}{x} v - v^2$. Page 147, Example #2: (Spotted by Momo Mizuguchi) Just before equation (23) we have: “Hence the general solution of the nonhomogeneous solution is”. The second “solution” should be “equation”. Page 148, Exercise #11: (Spotted by Abigail Jackson) In the last line, the right side of the equation should be$c_1 f_1(x) + c_2 f_2(x)$. (That is, the$y$'s should be changed to$f$'s.) Page 148, Exercise #13: (Spotted by Christine Sweigart, reported by Dr. Lo) The solution to part (b) in the back of the book is wrong. It should be: $y = c_1 x + \frac{c_2}{x} - 1.$ Page 150, Example #4: (Spotted by Dr. Gobbert) The expressions for$y_p$,$y'_p$and$y''_p$are correct, however in substituting them into the differential equation, the positions of$y_p$and$y''_p$have been swapped. Had this been done correctly, the final result in equation (13) would have been: $y_p(x) = \frac{3}{2} \cos 2x - \frac{1}{2} \sin 2x.$ Page 153, Table 3.2: (Comment by UHM) The entire Table 3.2 is equivalent to the following single rule: Consider the differential equation $ay'' + by' + cy = e^{\alpha x} \big[ P_n(x) \cos \beta x + Q_n(x) \sin \beta x \,\big],$ where$a$,$b$,$c$,$\alpha$and$\beta$are constants and$P_n(x)$and$Q_n(x)$are polynomials of up to$n\,$th degree for some$n \ge 0. Then a particular solution is: \begin{align*} y_p(x) = x^s e^{\alpha x} \big[ &(A_0 + A_1 x + \cdots + A_n x^n) \cos \beta x \\ +&(B_0 + B_1 x + \cdots + B_n x^n) \sin \beta x \,\big]. \end{align*} The exponents$is the multiplicity of$\alpha + i\beta$as a root of the characteristic polynomial$am^2 + bm + c$. Thus, if$\alpha + i\beta$is not a root, then$s=0$, if$\alpha + i\beta$is a root of multiplicity one then$s=1$, and if$\alpha + i\beta$is a root of multiplicity two then$s=2$. I suggest using this single rule instead of the nine rules in the textbook's Table 3.2. See the related comments in the following two headings. Page 154, Example #8: 1. (spotted by Celia Drew and reported by Dr. Dean) In the solution to part (d), the expression$f(x) = 2x\cos3x + \sin3x$should be$f(x) = 2x\sin3x + \cos3x$but this does not affect the rest of the solution. 2. (spotted by Celia Drew and reported by Dr. Dean) In the solution to part (e), the expression$f(x) = xe^{-2x}\cos x$should be$f(x) = xe^{-2x}\sin x$. Also, where it says “the terms of$f(x)$are of type 8” it should say “the terms of$f(x)$are of types 8 and 9” since the types 8 and 9” form an inseparable pair. 3. (Spotted by Momo Mizuguchi) In the solution to part (e), every$\sin3x$and$\cos3x$should be$\sin x$and$\cos x$. In any case, the 9 rules of the textbook's Table 3.2 make things more complicated than necessary. Instead, I suggest that you use the single rule formulated in the boxed comment above. See the following item for usage samples. Page 156, Exercises #31 and #35: (Spotted by Anthony Simms, reported by Dr. Peercy) The answers in the back of the book are wrong. The confusion is due to the less than ideal explanation of the 9 entries of the Table 3.2 on page 153. Here we solve these exercises by following the single rule formulated in the boxed comment above. • Exercise #31:$y'' + 4y = (x^2 + 2x -1) e^x \sin 2x$This corresponds to$\alpha=1$,$\beta=2$,$n=2$. Since$\alpha + i\beta$, that is$1 + 2i$, is different from the roots of the characteristic polynomial$m^2 + 4$(which are$\pm 2i$) then$s=0$and the particular solution takes the form: $y_p = e^x \big[ (A_0 + A_1 x + A_2 x^2) \cos 2x + (B_0 + B_1 x + B_2 x^2) \sin 2x \big].$ • Exercise #35:$y'' + 2y' + y = x^2 e^{-x} \sin 2x$This corresponds to$\alpha=-1$,$\beta=2$,$n=2$. Since$\alpha + i\beta$, that is$-1 + 2i$, is different from the roots of the characteristic polynomial$m^2 + 2m + 1$(which are$-1$and$-1$) then$s=0$and the particular solution takes the form: $y_p = e^{-x} \big[ (A_0 + A_1 x + A_2 x^2) \cos 2x + (B_0 + B_1 x + B_2 x^2) \sin 2x \big].$ Page 156, Exercise #7: (Spotted by Christine Sweigart, reported by Dr. Lo) The solution in the back of the book is not quite correct. It should be: $y = c_1 + c_2 e^{-4x} + \frac{1}{8} x^2 - \frac{1}{16} x.$ Page 158: The system of equations (9) and its solution in (10) correspond to the differential equation (1), namely $y'' + p(x) y' + q(x) y = f(x),$ where the coefficient of$y''$is 1. It would have been just as easy to work out the more general case where the coefficient of$y''is other than 1, as in: $a(x)y'' + b(x)y' + c(x)y = f(x).$ For this modified equation, the system of equations (9) would have changed to \begin{align*} y_1 v_1' + y_2 v_2' &= 0 \\ a(x) \big(y_1' v_1' + y_2' v_2'\big) &= f(x), \end{align*} and the corresponding solution in (10) would have been $v_1' = - \frac{y_2(x) f(x)}{a(x) W(y_1,y_2)} \quad\text{and}\quad v_2' = \frac{y_1(x) f(x)}{a(x) W(y_1,y_2)},$ whereW(y_1,y_2) = \det\big( \begin{smallmatrix} y_1 & y_2 \\ y'_1 & y'_2 \end{smallmatrix} \big) = y_1 y_2' - y_1' y_2$, as usual. I suggest using these instead of the textbook's equations (10). Page 159, the boxed Historical Note: (Spotted by Daniel Conlon, reported by Dr. Lo) It refers to Euler and Lagrange as seventeenth century mathematicians. Actually they were eighteenth century mathematicians. Page 160, Equation (12): (Spotted by Dr. Soane) Both integrals are indefinitie integrals (that is, antiderivatives) therefore naming the integration variables$t$is misleading and not quite correct. Change all occurrences of$t$to$x$. Page 161, Example 2: The solution is incorrect because the coefficient of$y''$in equation (19) is$x^2$, not 1, contrary to the assumption that was built into the preceding analysis. In fact, the differential equation, as given, cannot be solved in terms of elementary functions. To retain the solution given in the book, change equation (19) to $x^2 y'' - 2xy' + 2y = x^3 \sin x,$ then follow the method suggested in the boxed formula above. Page 162, Exercise #9: (Spotted by Dr. Bradley) The answer in the back of the book has a sign error. The term$-x \cos x$should be$+x \cos x$. Page 162, Exercise #10: (Spotted by Dr. Bradley) The answer in the back of the book has a sign error. The term$+x \cos x$should be$-x \cos x$. Page 162, Exercise #16: (Spotted by Dr. Bradley) The answer in the back of the book is given as$y_p = x^{1/2}$. It should be$y_p = x^{-1/2}$. Page 162, Exercise #17: (Spotted by Dr. Soane) The answer in the back of the book requires integration limits. It should be: $y_p(x) = \int_0^x \sinh(x-s) \, e^{-s^2} \, ds.$ Note: See Exercise 24 on page 163 for a similar case. Page 165, line 3: (Spotted by Andrew Waite) The second$\dot{u}$should be$\ddot{u}$. Page 166, Equation (3): (Spotted by Haley Boyd) The coefficient of$\dot{u}$should be 0.20, not 0.02. Page 167, Figure 3.9: Along the horizontal axis, the markings $\delta, \qquad \delta + \pi/\omega_0, \qquad \delta + 2\pi/\omega_0$ should be $\delta/\omega_0, \qquad (\delta + \pi)/\omega_0, \qquad (\delta + 2\pi)/\omega_0.$ Page 167: (Spotted by Dr. Bradley) In the second footnote, where it says Eq. (9), it should say Eq. (10). Furthermore, the expression$\tan\phi = c_2/c_1$should be$\tan\phi = c_1/c_2$. Page 168, near the bottom: (Spotted by Dr. Bradley) The equation$\tan\delta = \frac{c_1}{c_2} = -1$should be$\tan\delta = \frac{c_2}{c_1} = -1$. Page 168, near the bottom: We are told Since$c_1$is positive and$c_2$is negative, we see from Figure 3.10 that the phase angle$\delta$is in the second quadrant. Hence$\delta = 3\pi/4$. This is not correct. Since$c_1$is positive and$c_2$is negative, the phase angle$\delta$is in the fourth quadrant. Hence$\delta = -\pi/4$. Consequently, the equation (14) and the graphs on page 169 are incorrect. A related issue, spotted by Dr. Gobbert, is that the equation $\tan \delta = \frac{c_1}{c_2}$ near the bottom of page 168 is incorrect. The correct version is $\tan \delta = \frac{c_2}{c_1}$ which is obtained by dividing the expressions for$\sin\delta$and$\cos\delta$given in equation (9) on page 167. Page 170, Exercise #4: To get the answer in the back of the book, change the initial condition$\dot{u}(0) = 1$to$\dot{u}(0) = 3$. Page 170, Exercise #7: (Spotted by Christopher Gongora) In the back of the book, the$\dot{u}$in the answer to part (b) should be$\ddot{u}$. Furthermore, the two initial conditions there have run into each other. The proper form would be: $\ddot{u} + 64u = 0, \quad u(0)=-\frac{1}{6}, \quad \dot{u}(0) = 1.$ Page 170, Exercise #9: (Spotted by Dr. Dean) The answer in the back of the book is incorrect. It should be $u(t) = -\frac{1}{3} \cos 8t - \frac{1}{4} \sin 8t = \frac{5}{12} \cos(8t - \delta),$ where$\delta = \pi + \tan^{-1} \frac{3}{4} \approx 3.79$. Page 174, Figure 3.16: Along the horizontal axis, the markings $\delta, \qquad \delta + \pi/\mu, \qquad \delta + 2\pi/\mu, \qquad \delta + 3\pi/\mu$ should be $\delta/\mu, \qquad (\delta + \pi)/\mu, \qquad (\delta + 2\pi)/\mu, \qquad (\delta + 3\pi)/\mu.$ Page 179, Exercise #6: (Spotted by Dr. Dean) The equation has a typo; it's meant to be$\ddot{u} + 2\dot{u} + u = 0$. Page 179, Exercise #14: The numerical data as given leads to quite an unpleasant calculation. The answer in the back of the book is certainly wrong. Here is a restatement of the problem with data that leads to a tamer calculation. An 8-lb weight stretches a spring 8 inches, thereby reaching its equilibrium position. Assuming a damping constant for the system of 4lb/(ft/sec), the weight is pulled down by 3 inches below its equilibrium position and given a downward velocity of 3ft/sec. When will the mass attain its maximum displacement below equilibrium? What is maximum displacement? Sketch the graph of the motion. Answer:$ t = \frac{1}{8} \ln 2$. Page 180, Exercise #19: This exercise, involving forced vibrations, belongs to the next section. Pages 184 and 185, Example 1: Equations (10) and (11) on page 185, and the unnumbered equation that comes between them, need fixing. Suppose the road's surface is given by a function$h(x)$, where$x$is the coordinate along the road. If the car is at$x=0$when$t=0$, and travels at the constant speed$v$, then it is not too hard to see that the differential equation of motion is $$m \ddot{u} = -k \big[u(t) - h(vt)\big],$$ or equivalently, $m \ddot{u} + k u = k h(vt).$ In particular, since the problem's statement postulates that$h(x) = A \cos\big( \frac{2\pi x}{L} \big)$, then the differential equation of motion is $m \ddot{u} + k u = k A \cos\Big( \frac{2\pi vt}{L} \Big).$ This should replace the unnumbered equation that comes between equations (10) and (11). In view of this, equation (10) should be $f(t) = k A \cos\Big( \frac{2\pi vt}{L} \Big),$ and equation (11) should be $75 \ddot{u} + 7200 u = 7200 \cos\Big( \frac{2\pi vt}{25} \Big).$ Finally, near the lower right corner of Figure 3.24,$f(t)$should be$h(x)$. The rest of the solution is correct as written. Page 187, Exercise #5: (Spotted by Sam Phipps) The$t$in the answer in the back of the book should be$3t$, as in:$\ds u_{\text{st}} = \frac{4}{\sqrt{73}} \cos (3t - \delta)$. Page 187, Exercise #6: (Spotted by Dr. Bell) The solution to part (b) in the back of the book is wrong. Assuming zero initial conditions, it should be: $u = \frac{4\sqrt{3}}{3} t \sin\big(2\sqrt{3}t\big).$ Page 187, Exercise #8: (Spotted by Dr. Bell) The solution in the back of the book is wrong. The vertical motion of the buoy, measured from its water line (that's where the water surface contacts the buoy when the buoy is at rest) is given by $x(t) = A \cos \frac{2\pi}{5}t + B \sin \frac{2\pi}{5}t + \frac{25}{8} \sin \frac{2\pi}{7}t,$ where the arbitrary constants$A$and$B$depend on the initial conditions. The terms with$A$and$B$correspond to the buoy's natural oscillations. In reality they will go away after a while due to the water's damping effect (although this is not included in the model) and what will remain is the last term which corresponds to the forced motion of the buoy due to the action of the waves. The amplitude of the motion, 25/8, is slightly larger than 3 ft, therefore the buoy will be completely submerged in its lowest point. Pate 194, Figure 3.27: (An omission, not an error, spotted by UHM) The flowchart is incomplete—in the lower left corner, under “Roots$r_1,\ldots,r_n$”, the case of repeated complex roots is not accounted for. For completeness, the box dealing with complex roots should be replaced with: For each pair of complex roots$p+iq, p-iq$of multiplicity$k$, we obtain$2klinearly independent solutions \begin{align*} e^{px}\cos qx , \; x e^{px}\cos qx , \; \dots,\; x^{k-1} e^{px}\cos qx , \\ e^{px}\sin qx , \; x e^{px}\sin qx , \; \dots,\; x^{k-1} e^{px}\sin qx . \end{align*} Page 246, the boxed Historical Note: It refers to Laplace's “seminal treatise on probability, Mécanique Céleste”. The correct citation is Laplace's treatise on probability, Théorie analytique des probabilités, Paris, 1812. Aside: It is interesting to note that the Laplace transform had its origins in the theory of probability, not differential equations. An extensive historical account of the development of the Laplace transform is given in: 1. Michael A. B. Deakin, The Development of the Laplace Transform, 1737–1937. I. Euler to Spitzer, 1737–1880. Archive for history of exact sciences 25 (1981), no. 4, pp. 343–390. 2. Michael A. B. Deakin, The Development of the Laplace Transform, 1737–1937. II. Poincaré to Doetsch, 1880–1937. Archive for history of exact sciences 26 (1982), no. 4, pp. 351–381. Page 252, Exercise #27: (Spotted by Erica Ford) The1 \le t < 1$should be$0 \le t < 1$. Page 253, Exercise #46: (Spotted by UHM) In part (a), the denominator$1 - (a + ik)$should be$s - (a + ik)$. Page 254: (Spotted by Momo Mizuguchi) Under the Purpose paragraph, on the “derivative property” line, the expression$f\{0\}$should be$f(0)$. Page 255, Theorem 5.4 (boxed): (Spotted by Momo Mizuguchi) The term$s^{n-2} f''(0)$should be$s^{n-3} f''(0)$. Page 259, Exercise #7: (Spotted by UHM) The answer in the back of the book is off by a factor of 2. The correct answer is$\ds \frac{4(3s^2 - 4)}{(s^2+4)^3}$. Page 259, Exercise #12: (Spotted by Dr. Bell) The answer in the back of the book has a typo. The ‘4’ in the numerator should be ‘6’. Page 259, Exercise #17: (Spotted by Matthew Bleakney, reported by Dr. Sousedik) The answer in the back of the book has a typo—the sign between the two fractions should be a minus, not a plus. Page 259, Exercise #26: (Spotted by Momo Mizuguchi) The answer in the back of the book is off by a factor of 2. The correct answer is$\ds \frac{4(3s^2 - 4)}{(s^2+4)^3}$. Page 259, Exercise #42: (Spotted by Salen Nhean, reported by Dr. Sousedik) The upper bound in the inner integral should be$\tau$, not$u$. Page 266, Table 5.4: (Spotted by Momo Mizuguchi) In item 5, the large parentheses are meant to be large curly braces, as in$\mathscr{L}^{-1}\Big\{F(s-a)\Big\}$. Page 267, Exercise #12: (Spotted by Marina Congedo, reported by Dr. Hoffman) The answer in the back of the book is slightly off. It should be$\ds\frac{7}{\sqrt{3}} e^{-2t} \sin \sqrt{3} t$. Page 267, Exercise #28: (Spotted by Menachem Greenfeld) In parts (a) and (b) the functions$LC$and$RC$are referred to as “Laplace transforms”. However they should be referred to as inverse Laplace transforms. Page 270, lines 1 and 3 from below: (Spotted by Dr. Gobbert) The coefficients of$\sin 2t$on both of these lines should be$\frac13$rather than$\frac23. That is, \begin{align*} Y(s) &= \frac{1}{3(s^2+1)} + \frac{2}{3(s^2+4)} \\ &= \frac{1}{3} \mathscr{L}\{\sin t\} + \frac{1}{3} \mathscr{L}\{\sin 2t\} \end{align*} and $y(t) = \frac{1}{3} \sin t + \frac{1}{3} \sin 2t.$ (Additional comment from Dr. Sousedik) Consequently, the graph in Figure 5.7 does not represent the problem's correct solution. Page 271, Equation (9): (Spotted by UHM) The term3\big(\mathscr{L}\{y\} - y(0)\big)$in the middle of the left-hand side is missing an$s$. It should be$3\big(s\mathscr{L}\{y\} - y(0)\big)$. Page 273, Equation (17): There should be no minus sign in the exponent. The correct equation is: $Y(s) = \frac{1}{s^2} + \frac{C}{s^2} e^{s^2/2}.$ Page 274, Exercise #21: (Spotted by Pi Fisher) Equation (19) should be $\mathscr{L}\big\{ t^2 y'(t) \big\} = s Y''(s) + 2 Y'(s).$ Moreover, the wisdom of suggesting to apply Property 3 from Table 5.4 of page 266 is questionable. Applying Property 7 from Table 5.2 of page 258 makes better sense. Page 275, Exercise #26: (Spotted by UHM) Equation (26b) should be$(D+2)y = v$. Page 282, Exercise #3: (Spotted by UHM) The answer in the back of the book has a typo—the first minus should be a plus, as in: $u(t) + (4t - t^2 - 1)u(t-1) + (1 - 4t + t^2)u(t-2).$ Better yet, this may be expressed more clearly as $u(t) + (4t - t^2 - 1)u(t-1) - (4t - t^2 - 1)u(t-2).$ Page 284, sixth line from the bottom: (Spotted by Celia Drew, reported by Dr. Dean) The$\mathscr{L}\big\{ u(t-1) \big\}$should be$\mathscr{L}\big\{ u(t-\pi) \big\}$. Page 286, equation (7): (Spotted by Dr. Sousedik) There are two typos here: 1. In the equation's top line, the$e^t$should be$e^{-t}$. 2. In the equation's bottom line, there should be a minus sign in front of the$\frac{1}{2}$. Page 286, Example #3: (Spotted by Dr. Gobbert) The ‘1’ on the right-hand side of the equation should be ‘$\pi$’, as in: $y'' + 3y' + 2y = \begin{cases} t & (0 \le t < \pi) \\ \pi & (\pi \le t) \end{cases}$ Page 290, Exercise #4: (Spotted by Donna Young) The answer in the back of the book is not quite correct. It should be$ y(t) = t - \sin t - u(t-1)\big( t - \cos(t-1) - \sin(t-1) \big). $Page 290, Exercise #6: The solution in the back of the book is malformed. The correct solution is$y(t) = \sin t + u(t-3)\big[1 - \cos(t-3)\big]$. Page 290, Exercise #8: (Spotted by Dr. Bell) The answer in the back of the book has an extra$-u(t-\pi/2)$. It should have been $y(t) = \frac14 t - \frac18 \sin2t -u\big(t-\frac{\pi}{2}\big)\Big( \frac14\big(t-\frac{\pi}{2}\big) -\frac18\sin2\big(t-\frac{\pi}{2}\big) \Big).$ Page 290, Exercise #10: (Spotted by Christine Sweigart, reported by Dr. Lo) The answer in the back of the book is wrong. The correct solution is $y(t) = \frac{1}{3} \sin t - \frac{1}{6} \sin 2t - u(t-2\pi) \big[ \frac{1}{3} \sin (t-2\pi) - \frac{1}{6} \sin 2(t-2\pi) \big].$ Page 290, Exercise #11: (Spotted by Jeffrey Barden) The answer in the back of the book has an extra factor of 2 in one of its terms. Specifically, the$2\sqrt{3}$in the middle of the answer should be$\sqrt{3}$. Page 290, Exercise #12: (Spotted by Dr. Bell) The answer in the back of the book is incorrect. It should have been $y(t) = 2 - 2e^{-t}\cos2t - e^{-t}\sin2t - u(t-\pi) \Big( 2 - 2e^{-(t-\pi)}\cos2t - e^{-(t-\pi)} \sin2t \Big).$ This may be expressed more succinctly as $y(t) = g(t) - u(t-\pi) g(t-\pi), \quad\text{where } g(t) = 2 - 2e^{-t}\cos2t - e^{-t}\sin2t.$ Page 293, in the footnote: It says “The area under the curve has units of (force$\times$time) which has units of energy.” This is not correct. Energy would have units of (force$\times$distance). The integral of force with respect to time is called impulse in physics, not energy. Page 294: (Spotted by Menachem Greenfeld) The equation immediately above equation (2) is missing a factor of$m$on the left-hand side. Page 295, last paragraph: It says “Also, since$\delta(t)$represents an impulse of unit energy acting at$t=0$”. That should say “Also, since$\delta(t)$represents a unit impulse at$t=0$”. Energy and impulse are not the same things; see the previous remark. Page 296, Figure 5.32: (Spotted by Dr. Gobbert) In the diagram on the left, the height of the graph is shown as$\frac{1}{2}h$. It should be$\frac{1}{2h}$. Page 296: (Spotted by Dr. Gobbert) Two lines below the boxed Historical Note we have$\delta_h(t) = 1 - u(t-h)$. This should be$\delta_h(t) = \frac{1}{h}\big[1 - u(t-h)\big]$. Page 296, equation (4): (Spotted by Celia Drew, reported by Dr. Dean) In the final term the$e^{hs}$should be$e^{-hs}$. Page 297, Equation (5): (Spotted by Philip Ruijten) A factor of$s$is missing in the last fraction. The entire formula should be: $\mathscr{L}\big\{\delta(t)\big\} = \lim_{h\to0} \Big( \frac{1-e^{-hs}}{hs} \Big) = \lim_{h\to0} \Big( \frac{se^{-hs}}{s} \Big) = 1.$ Page 297, Equation (7): (Spotted by Dr. Gobbert) We have: $\delta_h(t) = \frac{1}{h}\big[1 - u(1-h)\big]$ This should be $\delta_h(t) = \frac{1}{h}\big[1 - u(t-h)\big]$ Page 297, last line: It says “it is struck by a hammer in the downward direction, exerting unit energy on the mass”. That “unit energy” should be “unit impulse”. Energy and impulse are not the same things; see the preceding remarks on this issue. Page 301, Exercise #11: (Spotted by Momo Mizuguchi) In the answer in the back of the book, the term$2\sin t$should be just$\sin t$. Page 311, Exercise #17: (Spotted by UHM) The answer in the back of the book is not quite correct. It should be$y = \frac12 \big[ \sin 2t + \sin 2t \ast f(t) \big]$. Page 311, Exercise #19: (Spotted by Dr. Bell) The answer in the back of the book has a typo. The term$4\sin t$in it should be$2\sin t$. Page 311, Exercise #23: (Spotted by Bradley Scott) The answer in the back of the book has a typo. It should be$4t \ast e^{2t} = e^{2t} - 2t - 1$. Page 316: (Spotted by Dr. Sousedik) Within the shaded box, all 9 occurrences of$x_0$should be$t_0$, Page 319, equation immediately preceding equation (17): (Spotted by Menachem Greenfeld) On the right-hand side, the term$x_2 c_1 e^{3t}$should be$c_1 e^{3t}$. Equation (17), however, is correct. Page 344, Figure 6.3: (Spotted by Momo Mizuguchi) The point labeled$(-1,2)$should be labeled$(1,-2)$. Page 353, Exercise #27: (Spotted by Dr. Bell) In part (b), the condition$(a-d)^2=0$should be$(a-d)^2 + 4bc=0$. Page 357, Equation (14): The expression$(1-\lambda)(\lambda^2+2)$should be$(1-\lambda)(\lambda^2+4)$. Page 368, last line before the footnote: It says: “The following example illustrates that nonlinear equations…”. The “nonlinear” should be “nonhomogeneous”. Pages 376–378, Exampe 2: (Spotted by Thomas Hsu) A numerical error has crept in Equation (4) on page 377—the eigenvectors are incorrect as shown. The corrected version of the equation is: $x_h(t) = \begin{bmatrix} x_1(t)\\x_2(t) \end{bmatrix} = c_1 e^{-0.025 t} \begin{bmatrix} 1 \\ 1.732 \end{bmatrix} + c_2 e^{-0.095 t} \begin{bmatrix} 1 \\ -1.732 \end{bmatrix}.$ Equation (5) on page 378 needs to be corrected accordingly. Upon applying the initial conditions we find that$c_1 = -26.93$and$c_2 = 1.93$. Finally, we redo the graphs in Figure 6.13 on page 378 and we get: Aside: If instead of decimal points we use true fractions in the computations, we see that the eigenvalues and eigenvectors, and the coefficients$c_1$and$c_2$are: $\lambda = \frac{1}{50} \big[ -3 \pm \sqrt{3} \big], \quad \mathbf{v} = \begin{bmatrix} 1 \\ \pm \sqrt{3} \end{bmatrix}, \quad c_1 = -25\Big(\frac{1}{2} + \frac{1}{3} \sqrt{3}\Big), \quad c_2 = -25\Big(\frac{1}{2} - \frac{1}{3} \sqrt{3}\Big).$ Page 397: (Spotted by Menachem Greenfeld) Top of the page, equation (b): The first$y_{n+1}$should be$y_{n+2}$. Pages 475–476: (Spotted by Dr. Neofytos Komninos) Near the top of page 476 we have: “Clearly, the low state is asymptotically stable…”. This is in reference to the motion of the undamped pendulum (equation (10) on page 475). The correct statement should be: “Clearly, the low state is stable but not asymptotically stable”. Further comment: That erronous statement has crept in possibly under the influence of the nonsensical illustration in Figure 8.27 on page 475. A static figure like that cannot convey the concept of asymptotic stability as it is attempting to do. Ignore it. Pages 479, equations (15) and (16): The discussion under Linearizing Nonliner Systems introduces the linearized system (16) corresponding to an equilibrium point$(x_0, y_0)$of the nonlinear system (15). Then it makes the statement: “The trajectories of the linearized system approximate the trajectories of the nonlinear system (15) in a small region near$(\xi, \eta)$.” There are two errors in that statement. • A minor error: The “small region near$(\xi, \eta)$” should be “small region near$(x_0,y_0)$”. • A major error: The statement is guaranteed to be true if the linearized system (16) has no eigenvalues on the imaginary axis in the complex plane. Otherwise, the linearized system provides no information about the nonlinear system. This is the subject of the celebrated Hartman–Grobman Theorem. It is alarming how often that essential “fine print” on eigenvalues is omitted/ignored in published articles and books, especially in the engineering literature. Be on guard! To the book's credit, the linearlization theorem is revisited in Section 8.3 where boxes on page 483 correctly state the linearization theorem and point out the need for the assumption on the eigenvalues. Pages 479, Exercise #18: (Spotted by Dr. Neofytos Komninos) The answer in the back of the book says$\dot{x}=y$. It should say$\dot{x}=x$. Pages 479, Exercise #19: The linearized system has eigenvalues$\pm i$. Since these lie on the imaginary axis in the complex plane, the linearization theorem provides no information about stability—see the previous comment with referece to the Hartman–Grobman Theorem. The answer in the back of the book is correct, however it does not follow from the linearization theorem. It has to be proved through independent means. Pages 479, Exercise #20: The answer in the back of the book gives the linearized equation correctly, but its analysis of the cases is incorrect. Furthermore, the equation$\dot{y} = \epsilon(x^2-1)y - x$is not Van der Pol's equation, despite the problem's statement. Van der Pol's equation would be$\dot{y} = -\epsilon(x^2-1)y - x$. Pages 484, Example 2: (Spotted by Dr. Neofytos Komninos) The analysis is correct up to equation (10) but the conclusion that the eigenvalues are$\lambda_1=-1$and$\lambda_2=3$is incorrect. The correct eigenvalues are$1 \pm i\sqrt{2}$, and consequently the equilibrium is an unstable spiral. Page 493, the boxed Historical Note: It refers to “The Soviet mathematician Aleksandr Mikhailovich Liapunov (1857–1918)…”. However Liapunov was not a Soviet mathematician—he died before there was a Soviet Union. As the footnote on page 488 correctly states, Liapunov was a Russian mathematician. Page 569: (Spotted by Menachem Greenfeld) In the next to last step in equation (5), the denominator is$c^2 - cdi + cdi - d^2i$but it should be$c^2 - cdi + cdi - d^2i^2\$.
Inside back cover:
(Spotted by Menachem Greenfeld) There is an extraneous parenthesis in formula 17. The correct expression is: $\int \sqrt{u^2 \pm a^2} \, du = \frac{1}{2} \Big( u \sqrt{u^2 \pm a^2} \pm a^2 \ln \big( u + \sqrt{u^2 \pm a^2} \, \big) \Big)$

 Author: Rouben Rostamian