Section 1.3, Page 44, Figure 1.28

In the text accompanying Figure 1.28, replace K = 1 with K = 2.

Section 1.3, Page 45, Exercise 1.
The answer in the back of the book is wrong. The sketch provided there is apparently for

Section 1.6, Page 84. The displayed equation in the middle of the page
dS/dt = kS (1 - S/M) (S/M - 1) should read

Section 1.6, Page 85. In the caption of Figure 1.75, the displayed equation should read

Section 1.7, Page 94. The beginning of the second paragraph should read:

First, if alpha < 0, the term ...

Also, the caption in Figure 1.82 is incorrect. The second sentence should read:

Note that for alpha < or = 0 the graph crosses the y-axis once, whereas if alpha > 0, the graph crosses the y-axis three times.

Section 1.7, Page 97, Figures 1.86 and 1.87. These figures do not correspond well with the text that describes them on pages 96 and 97. In Figure 1.86, the bottom curve should have three real roots rather than just one implying three equilibrium points. But this only happens for -4/27 < mu < 0. We chose a mu value too negative or too far away from the bifurcation point mu = 0!

In Figure 1.87, the top curve should have three real roots rather than just one implying three equilibrium points. Again, this happens for -4/27 < mu < 0. When graphing the top curve, we overshot the value mu = 0, and actually chose a positive mu value!

The moral of this story is that bifurcations should be analyzed locally. When a bifurcation happens at say mu = 0, then one should really only look at graphs with mu values very close to 0. We failed to do this and obtained misleading graphs.

Section 1.7, Page 100, Exercise 3. The answer in the back of the book is incomplete. In addition to the bifurcation at a = 2, there is a similar bifurcation at a = -2. Here, the phase lines for a > -2, a = -2 and a < -2 are qualitatively the same as the ones for a < 2, a = 2 and a > 2 respectively.

Section 1.8, Page 113, Exercise 25. The answer in the back of the book is incorrect. In our haste, we failed to express the amount of dioxin as a concentration. The correct answer is

Note that starting with a concentration of 2 ppb and adding water with a concentration of 5 ppb will result in a concentration somewhere between 2 and 5, nowhere near 1700! (Our grad students must have been sleeping on that one.) Although the problem is solvable without converting ppb by weight to lbs/gal, we recommend this approach to clarify the problem.

Section 1.9, Page 125, Exercise 15. This problem and the answer in the back of the book are messed up. As stated in the problem, the volume in the tank remains constant, but the answer in the back of the book states that the volume is given by 10 + 3t. We suggest replacing the problem with the following new problem:

Consider a very large vat that initially contains 10 gallons of clean water. Suppose water starts entering the vat from two pipes. From the first pipe, saltwater containing 2 pounds of salt per gallon enters the vat at a rate of 1 gallon per minute. From the second pipe, saltwater containing 0.2 pounds of salt per gallon enters the vat at a rate of 5 gallons per minute. Suppose the liquid is kept well mixed and saltwater is removed from the vat at a rate of 3 gallons per minute.

• Derive a differential equation for the rate of change of the total pounds of salt S(t) in the vat at time t.

• Convert this equation to a differential equation for the concentration C(t) of salt in the vat at time t.

• What will the concentration of salt in the vat be after a very long time?

• Find the concentration of salt in the vat at the time t=5.

With these modifications, the answer in the back of the book is now correct.

Lab 1.1, Page 129.
The astute observer and well-traveled person might have noticed that Alaska has nowhere near 4 million people in its current population. The list of populations for Alaska is really that of Alabama . Sorry for offending all you "Northern Exposure" fans.

Section 2.4, Page 183, Exercise 13.
There is a slight error in the back of the book. In the equation for dv/dt, the correct coefficient for the y term is -(k_1 + k_2)/m.

Lab 2.3, Page 221.
There is an error in the first table. The first entry in the column for y_2(t) should be 0.74, not .074.

Section 3.2, Page 256.
In the Theorem at the top of the page, when there are distinct, real eigenvalues lambda_1 and lambda_2, the solutions are

Section 3.2, Page 261, Exercises 23-28.
The instructions next to the third bullet are vague. Feel free to omit these when doing problems 23-28.

Section 3.2, Page 261, Exercise 27.
There is a typo in the answer to part c in the back of the book (p. 621). The term 2 + sqrt{33} should be replaced by simply 2 sqrt{33}.

Section 3.3, Page 276, Exercise 15d.
The y(t)-graph in the answer in the back of the book is wrong. We'll put the correct one up here soon. Also, although the answer provides a v(t)-graph, this was not called for in the problem. Nice of us to give you extra info, ehh?

Section 3.4, Page 290, Exercise 11a.
The answer in the back of the book is wrong. The e^{4t} term should be placed between k_2 and the given vector.

Section 3.4, Page 291, Exercise 15.
The answer in the back of the book asserts that graphs 2 and 5 are correct, and that the natural period of graph 1 is not constant over time. This is somewhat difficult to see, so one might argue that graph 1 is also acceptable as the graph of the x-coordinate of a solution.
Also, in the exercise, graphs 1, 4, and 6 have axes labled y instead of x. This is a typo.

Section 3.4, Page 292, Exercise 22.
The hint given in the statement of the problem is incorrect; k_1 and k_2 should be interchanged.

Section 3.4, Page 292, Exercise 25.
The answer in the back of the book is incorrect. Part (c) should not have a graph at all. It should say underdamped . Part (d) should have the graph from part (c). Part (e), the general solution, should read:

Section 3.6, Page 316.
As it reads, we have an "imaginary" period for our underdamped oscillator at the top of the page. The period 4 PI/ sqrt{K_d^2 - 4} should be replaced by

Section 3.6, Page 320, Exercise 9.
The answer in the back of the book is incorrect. Part (a) should read:

Part (b) should read:

Section 3.6, Page 321, Exercise 27.
The answer in the back of the book is incorrect. In parts (a) and (b), there should be sqrt signs over 3t/2 and 3/2 everywhere they occur in the answers. Moreover, in part (b) the coefficients of the sin terms are incorrect. Both coefficients should be replaced by -sqrt{6}.

Section 3.6, Page 323, Exercise 35.
The answer in the back of the book is incorrect. Part(a) should read:

and

Part (b) should be adjusted to: Modify the part of this section entitled "The Harmonic Oscillator" to consider the change in the sign of the dy/dt term.

Also, on the following page, the eigenvalues in the first line should be 0.1 + i and 0.1 - i.

Section 3.7, Page 335,
The authors have a little trouble with decimals every now and then. In lines 10 and 12, the eigenvalue should be lambda_1 = 0.2 .

The equations for the eigenvector for lambda_1 = 0.2 should be

The equations for the eigenvector for lambda_2 = -0.1 + isqrt{0.03} should be

Section 3.7, Page 338, Exercise 11a.
Somehow a negative sign emerged in the answer. The correct eigenvalues are -2 and 1, not -2 and -1.

Section 4.1, Page 359, Exercise 2
The third system in this problem actually doesn't have an equilibrium point at the origin. Change the second equation in this system to read dy/dt = 4x + 3 cos y - 3.

Section 4.2, Page 373
We have inadvertantly introduced the variable y here in the text when we really meant to use theta. This occurs in the text on this page three times. The caption should also be theta(t) instead of y(t).

Section 4.3, Page 391
The three references to y(0) in the second paragraph should be x(0) .

Section 4.4, Page 400
The characteristic polynomial is incorrect, it should read:

Don't know how this error crept into the book since the right answer appears several times in the section!

That should get you going in the right direction!

Section 8.2, Page 577, Exercise 17.
There is a typo in the answer in the back of the book. The correct answer is: