Errata in Farlow's Differential Equations — Chapter 7

Page 397:

(Spotted by Menachem Greenfeld) Top of the page, equation (b): The first $y_{n+1}$ should be $y_{n+2}$.

Page 408, Exercise 6:

(Spotted by Paul Rydeen) The equation $y_{n+2} - y_{n} - 6 y_n = 0$ is meant to be $y_{n+2} - y_{n+1} - 6 y_n = 0$.

Page 416, Example 6:

(Spotted by Paul Rydeen) The equation at the bottom of that page is missing the $-z y_0$ term. Later on $y_0$ is set to zero, therefore this does not affect the rest of the computation

Page 420, Exercise 27:

(Spotted by Paul Rydeen) A factor of $t$ is missing in the book's answer. The correct answer is $\frac{az}{z - e^{-kt}}$.

Page 420, Exercise 35:

(Spotted by Paul Rydeen) The answer in the back of the book has an extra zero. The correct answer is $\Bigl\{ -\frac{1}{2}, -\frac{1}{4}, -\frac{1}{8},\ldots \Bigr\}$.

Page 420, Exercise 37:

(Spotted by Paul Rydeen) The answer in the back of the book is incorrect. It should be $\Bigl\{ 1, 0, \frac{1}{2^2}, 0, \frac{1}{2^4}, 0, \ldots \Bigr\}$.

Page 422, Example 1:

(Spotted by Paul Rydeen) The solution should end in $.95$ rather than $.90$.

Page 426, Exercise 9:

(Spotted by Paul Rydeen) The book's answer to part (a) is correct, but those of parts (b) and (c) are incorrect since those are calculated based on $S_0 = 1000$. The correct initial condition is $S_1=1000$, and therefore the solution of the difference equation is $S_n = 20000 (1.1)^n - 20000 (1.05)^n$. Consequently, $S_4 = 4971.87$ and $S_{50} = 2,118,469.06$.

Page 428, Exercise 21:

(Spotted by Paul Rydeen) The question “how many distinct regions are formed” should be “what is the maximum number of the distinct regions formed” since the number of regions can be smaller if more than two lines are concurrent.

Moreover, the outlined idea of the proof, where it says “the $(n+1)$st line … will divide the previously constructed regions into twice that number” is either incorrect or not useful, depending on the interpretation, although the difference equation given at the end of the problem's statement is correct. For the proper reasoning behind the derivation of that equation see Number of Regions N Lines Divide Plane.

Page 438, Exercise 20:

(Spotted by Paul Rydeen) The book says that the behavior is chaotic in both cases, but it appears that the behavior in part (b) depends on the initial data. For instance, with $x_0=0.2$ we get a 2-cycle $0.2, 0.8, 0.2, 0.8,\ldots$, while $x_0=0.21$ leads to a cycle of length 10 consisting of $0.16, 0.32, 0.64, 0.72, 0.56, 0.88, 0.24, 0.48, 0.96, 0.08$.

Further numerical experiments show that $x_0=0.234$ leads to a cycle of length 49. It is not clear whether any initial condition leads to a truly chaotic behavior.

Page 439, Exercise 21(c):

The book's answer has $\lim_{x\to\infty}$. That should be $\lim_{n\to\infty}$.