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UMBC Dept of Math & Stat

Errata in Farlow's Differential Equations — Chapter 6

Page 316:
(Spotted by Dr. Sousedik) Within the shaded box, all 9 occurrences of x0 should be t0,
Page 319, equation immediately preceding equation (17):
(Spotted by Menachem Greenfeld) On the right-hand side, the term x2c1e3t should be c1e3t. Equation (17), however, is correct.
Page 323, Exercise #24:
(Spotted by Ethan Piercy, reported by Dr. Peercy) The answer for r(t) in the back of the book is correct, but j(t) has its sign reversed. It should be j(t)=c1bacos(abt)+c2basin(abt)
Page 330, equation (20):
(Spotted by Noah Zambidis) The ddx should be ddt.
Page 338, line 2:
(Spotted by Ryan Olson) xij(x) should be xij(t).
Page 344, Figure 6.3:
(Spotted by Momo Mizuguchi) The point labeled (1,2) should be labeled (1,2).
Pages 349–350, Example 5:
(Spotted by Paul Rydeen) In equation (28) the 2 and 2 have been swapped. The correct version is \boldsymbol{\xi}^{(1)} = c \begin{bmatrix} 1 \\ -2 \end{bmatrix} \qquad \boldsymbol{\xi}^{(2)} = c \begin{bmatrix} 1 \\ 2 \end{bmatrix} The 2 and -2 should be swapped in the same way in equations (29a), (29b), and (30).
Page 352, Exercise #18:
(Spotted by Paul Rydeen) In the answer for x_2(t) in the back of the book, e^{2t} should be e^{3t}.
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 353, Exercise #30:
(Spotted by Paul Rydeen) The answer to part (b) in the back of the book in incorrect. It should be \begin{bmatrix} I(t) \\ X(t) \end{bmatrix} = c_1 e^{-k_1 t} \begin{bmatrix} k_2 - k_1 \\ k_1 \end{bmatrix} + c_2 e^{-k_2 t} \begin{bmatrix} 0 \\ 1 \end{bmatrix}
Page 357, Equation (14):
The expression (1-\lambda)(\lambda^2+2) should be (1-\lambda)(\lambda^2+4).
Page 358–359, Exercises #13, 14, 18:
(Spotted by Paul Rydeen) The solution to each of these exercises should be of the form given in equation (18).The answers in the back of the book are missing the c multiplier.
Page 358, Exercise #15:
(Spotted by Paul Rydeen) In the comments preceding Exercise 15, the vectors x^{(1)}(t) and x^{(2)}(t), and the matrix X(t) should be x^{(1)}(t) = \begin{bmatrix} e^{-t} \\ -2 e^{-t} \end{bmatrix}, \quad x^{(2)}(t) = \begin{bmatrix} e^{3t} \\ 2 e^{3t} \end{bmatrix}, \quad X(t) = \begin{bmatrix} e^{-t} & e^{3t} \\ -2 e^{-t} & 2e^{3t} \end{bmatrix}.
Page 361, Example 1:
(Spotted by Matthew Lastner) In line 1 of equation (7), the matrix marked A^4 should be the identity matrix, that is: A^4 \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. The rest of the calculation is correct as is.
Page 364, in the boxed statement of Theorem 6.9:
(Spotted by Paul Rydeen) The general solution is given as \dot{x}. If should be x without a dot.
Page 366, Exercise #5:
(Spotted by Paul Rydeen) The solution in the back of the book needs to be fixed. The last term of the solution which is given as
-\left[\begin{array}{r} -10 \\ 1 \end{array}\right],
should be
-\left[\begin{array}{r} \frac{14}{3} \\ -\frac{11}{3} \end{array}\right].
Page 366, Exercise #7:
(Spotted by Paul Rydeen) The solution in the back of the book needs to be fixed. The last term of the solution which is given as \frac{1}{2}\left[\begin{array}{r} e^{5t} \\ e^{5t} \end{array}\right], should be \left[\begin{array}{r} 0 \\ -e^{5t} \end{array}\right], or equivalently, e^{5t} \! \left[\begin{array}{r} 0 \\ -1 \end{array}\right].
Page 366, Exercise #11:
(Spotted by Paul Rydeen) The right-hand side of the equation should have e^{-At} rather than e^{At}.
Page 368, last line before the footnote:
It says: “The following example illustrates that nonlinear equations…”. The “nonlinear” should be “nonhomogeneous”.
Page 369, Example 2, Equation (8):
(Spotted by Paul Rydeen) X_1(x) should be X_1(s).
Page 371, Example 3, Equation (14):
(Spotted by Paul Rydeen) In the second equation of the system (14), the -m_1 should be m_1.
Page 373, Exercise #8:
(Spotted by Paul Rydeen) The answer in the back of the book says x(t) = \cdots. It should say \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \cdots.
Page 374, Exercise #16:
(Spotted by Paul Rydeen) In the first equation, the term k_1m should be -k_1m.
Pages 376–378, Example 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:
[graph]

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 379, Equation (8):
(Spotted by Paul Rydeen) The m_1 in the second line should be m_2.
Page 382, Equation (12):
(Spotted by Paul Rydeen) That equation is incorrect since it accounts for material entering each compartment but does not account for material leaving that compartment. The correct equation is \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} 0 &\quad a_{12} - a_{21} &\quad a_{13} - a_{31} \\ a_{21} - a_{12} &\quad 0 &\quad a_{23} - a_{32} \\ a_{31} - a_{13} &\quad a_{32} - a_{23} &\quad 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}.
Page 383, Exercises #6:
(Spotted by Paul Rydeen) In view of the comment above, the correct answer to part (a) is \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} 0 & -0.02 & -0.06 \\ 0.02 & 0 & 0.04 \\ 0.06 & -0.04 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}.
Page 383, Exercises #7:
(Spotted by Paul Rydeen) As in the previous comment, the correct system of differential equations is \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} 0 & -0.25 & 0.15 \\ 0.25 & 0 & -0.2 \\ -0.15 & 0.2 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}.
Pages 386–387, Example 1:
(Spotted by Paul Rydeen) The first two lines of equation (9b) are correct but in the third line the final parenthesized 1 should be 2. On the next line, the answer -0.1 should be changed to -0.099666\ldots. That value is shown correctly in Table 6.1.
Page 389, Example 2:
(Spotted by Paul Rydeen) In the second line of the Solution, the x_0=1 should be x_0=2. The rest of the calculations uses the correct value of x_0=2.
Page 391, Exercises 4–8:
(Spotted by Paul Rydeen) Following the notation of Example 1, we introduce x_1=y, x_2=y' (and in the case of Exercise 8, x_3 = y''), so the solutions would be things like x_1(0.1) = \text{something}, x_2(0.1) = \text{something}, etc. The answers given in the back of the book have x(0.1) = \text{something}, y(0.1) = \text{something} which are inconsistent with that notation.
Page 391, Exercise 8:
(Spotted by Paul Rydeen) The book's answer is correct but incomplete. To covert this third order equation into a system, we let x_1 = y, x_2 = y', x_3 = y''. Then we get \begin{align*} x_1' &= x_2, \\ x_2' &= x_3, \\ x_3' &= - x_1^2, \end{align*} along with the initial conditions x_1(0)=1, x_2(0) = 0, x_3(0) = 0. Applying the Runge--Kutta algorithm with h=1/10 results in x_1\bigl(\frac{1}{10}\bigr) = \frac{5999}{6000}, \quad x_2\bigl(\frac{1}{10}\bigr) = -\frac{1}{200}, \quad x_3\bigl(\frac{1}{10}\bigr) = -\frac{95992001}{960000000}, or expressed in decimal fractions x_1(0.1) \approx 0.9998333333, \quad x_2(0.1) \approx -0.005, \quad x_3(0.1) \approx -0.0999916677.



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Author: Rouben Rostamian