Bill Goodwine's Courses

Posted: **Tue Aug 26, 2008 2:09 pm**

You may download the updated version of the course text. This may contain additional material and corrections that are not in the printed version. Unless otherwise indicated in class or on a homework, all references to page, section and problem numbers are for the printed version.

Post any errors that you find in the printed version of the course text below. To aid in avoiding any repeats, please do it as follows:

In the subject put something like

"page xx, line yy"

then in the body of the post put the error. If it's near the bottom of the page put "page xx, line -yy," meaning the error is yy lines up from the bottom. Count equations as lines as well.

Points:

typos = 1 homework point
substantive errors = 10 homework points
really major errors = 10's of homework points

Obviously what constitutes the distinction between these three types of errors is a matter of degree. I'll use my best judgment to allocate points fairly.

Only the first person to post an error will receive any credit, so try not to repeat any others that have already been posted.

Posted: **Sat Aug 30, 2008 12:15 pm**

"that" is repetitive (2nd word of the line) and a space is needed between "and" and "1.6.4"

Posted: **Sat Aug 30, 2008 2:31 pm**

Page 16, Heading 1.8: There's nothing written under the "Stability" heading. (Same on page 62, heading 2.4, page 88, heading 3.5, and on page 161, heading 6.8)
Page 42, Problem 6: The 3rd line in the problem is misaligned.

Posted: **Sun Aug 31, 2008 8:34 am**

Possibly missing article(s)?

"In this book, a relation between elements of sets may be defined by [an] equation or [a] set of equations."

Posted: **Sun Aug 31, 2008 9:31 am**

Missing pronoun?

"For all the general solutions we will consider in this book[, this/it] is, in fact, the case."

Posted: **Sun Aug 31, 2008 10:58 am**

Sorry, Professor, that second typo is actually on page 16, line 2 of the "updated version" of the text:

"For all the general solutions we will consider in this book[, this/it] is, in fact, the case."

Posted: **Tue Sep 02, 2008 10:19 am**

5. The letter i usually represents the imaginary unit, which satisfies i^2 = -1. The letter i may also be used to represent a function of time representing electric current, so is usually denoted i(t).

I believe it should read:

5. The letter i usually represents the imaginary unit, which satisfies i^2 = -1. The letter i may also be used to represent a function of time representing electric current, so it is usually denoted i(t).

"it" was added after the comma in sentence 2.

Posted: **Thu Sep 04, 2008 3:49 pm**

only one "that" is necessary

"Observe that that in Examples 1.6.2 and 1.6.4 there are arbitrary constants in the solution. We
will also be interested in the case where there are no arbitrary constants."

-Steven Rivera

Posted: **Thu Sep 04, 2008 3:52 pm**

believe it should be: "[we] will consider"

"For the types of differential equations will consider in..."

-Steven Rivera

Posted: **Thu Sep 04, 2008 3:55 pm**

believe "for" should be removed

"Now, for a rigid body is simply a system of particles where the particles are constrained by internal
forces to remain a fixed distance from each other."

-Steven Rivera

Posted: **Thu Sep 04, 2008 6:35 pm**

typo:

"For all the general solutions we will consider in this book _it_ is, in fact, the case."

Posted: **Fri Sep 05, 2008 9:16 am**

Page 25, line 12
"In the next section will consider the special case where the system...."

Shouldn't the "In" be removed or the "will" be changed to "we'll"?

Posted: **Fri Sep 05, 2008 10:26 pm**

After Example 2.2.1, I'm guessing the word "paced" should be "placed."

Posted: **Sat Sep 06, 2008 11:14 am**

reads: "In this example... f(x,t) = -3x*e^(sin2t)."

In Euler's Method, f(x,t) should equal the given dx/dt, right? So in this case it would simply be:

"f(x,t) = sin2t."

This (the sin2t) seems to be what you used for variable "f" in the program you put in the appendix as well.

Posted: **Sat Sep 06, 2008 11:34 am**

reads: "x_2(dot) = f_1(x_2(t),x_2(t),t)."

Should this be " = f_2(x_1(t),x_2(t),t." ?
Looks like the subscripts got swapped.

Posted: **Mon Sep 08, 2008 11:16 am**

"system of differential equation" should be "system of fifferential equations" in italics

Posted: **Mon Sep 08, 2008 11:17 am**

preposition "to" is repeated : ". . . no way to just to be able . . ."

Posted: **Mon Sep 08, 2008 11:30 am**

should

h = r x v

be

h = m (r x v)

as in line 11 above?

Posted: **Mon Sep 08, 2008 11:31 am**

"constrain" should be "contraint"

Posted: **Mon Sep 08, 2008 11:32 am**

"schematics" should be "schematic"

Posted: **Mon Sep 08, 2008 2:07 pm**

punctuation: period missing after sentence ending with ". . . gain^12" before sentence beginning with "The mathematical description is . . ."

Posted: **Mon Sep 08, 2008 2:11 pm**

formatting: section of text beginning with "There is something . . ." has words cut off at the end of each line and is unreadable

Posted: **Mon Sep 08, 2008 2:16 pm**

punctuation: commas missing around equations in sentence after "Mathematically" and before "where . . ."

compare to similar sentence on page 27, lines 11-12

Posted: **Tue Sep 09, 2008 2:14 pm**

If the solution indicated by that first equation

x(t) = (15/13)e^-3t + (-3/13)cos(2t) + (3/13)sin(2t)

is a continuation from the previous page, the -3/13 should be a -2/13.

Posted: **Tue Sep 09, 2008 5:10 pm**

page 542 (program for example 1.10.2)

"To compile... type 'f77 eulerexample.f'" (not "exulerexample")

Posted: **Tue Sep 09, 2008 5:48 pm**

Page 541, line 05

"This is a sampe FORTRAN program..." sampe should be sample

Page 542, line 03

"This is a sampe FORTRAN program..." sampe should be sample

Page 543, line 03

"This is a sampe FORTRAN program..." sampe should be sample

Page 548, line 4

"This is a sampe FORTRAN program..." sampe should be sample

Page 549, line 4

"This is a sample FORTRAN program..." sampe should sample

Page 550, line 4

"This is a sample FORTRAN program..." sampe should sample

Page 551, line 4

"This is a sample FORTRAN program..." sampe should sample

Posted: **Tue Sep 09, 2008 10:12 pm**

"in" should be "is"

Posted: **Tue Sep 09, 2008 10:14 pm**

In example 2.2.1 on page 46 it should "the temperature of an object PLACED in an oven" it presently says "paced".

Posted: **Wed Sep 10, 2008 4:50 pm**

"Given that, the author feels that fundamental theoretical understanding of the subject matter is becoming more important that (<- should be "than") proficiency in (hand) solution techniques."

Posted: **Sun Sep 14, 2008 12:27 pm**

There is an extra space before the period on line 12. It reads ". . . is the imaginary part ."

And I do realize that isn't a very exciting typo to point out.

Posted: **Sun Sep 14, 2008 12:33 pm**

Line reads currently ". . .relate the Cartesian for to polar form."

Believe it should read ". . .relate the Cartesian form to polar form."

Posted: **Sun Sep 14, 2008 4:32 pm**

The text currently reads "At this point is suffices to note that..."

I think it is supposed to say "At this point it suffices to note that..."

Posted: **Sun Sep 14, 2008 4:40 pm**

I think this is another one.
Text line 3 currently reads "order equation (equation 3.12) has been with another one (equation 3.14)..."

I think it should be "order equation (equation 3.12) has been replaced with another one (equation 3.14)..."

Posted: **Sun Sep 14, 2008 7:58 pm**

"and substituting gives
x** + 5* + 6x = (lambda^2)(e^lambda*t) + 5lambda(e^lambda*t) + e^lambda*t"
i think the last term should read 6*e^(lambda*t). The 6 was left off

Posted: **Sun Sep 14, 2008 10:24 pm**

the word "coefficients" is not capitalized and the word "constant" is spelled incorrectly. the title should instead read "The Method of Undetermined Coefficients for Constant Coefficient Equations"

Posted: **Sun Sep 14, 2008 11:04 pm**

"...that contains the product of the all the corresponding linearly..."
the "the" before the all should be cut.

Posted: **Mon Sep 15, 2008 2:36 pm**

The following equation in line 18 has an error.
x'' + omega_n * x = (F/m)*cos(omega*t)

As omega_n = (k/m)^1/2, this term must be squared to avoid altering the equation. It should read:

x'' + (omega_n)^2 * x = (F/m)*cos(omega*t)

Posted: **Mon Sep 15, 2008 8:49 pm**

Definition 3.2.7 states "... is called linearly independent ..."

It should be "linearly dependent"

Posted: **Mon Sep 15, 2008 8:53 pm**

The final line of the definition of the damped natural frequency states:

Wd = sqrt[ (4mk-b^2) / 2m )

it should be

= [sqrt (4mk-b^2) ] / 2m

That is, the sqrt of the denominator (4m^2) is erroneously declared = sqrt(2m) rather than just 2m

Posted: **Mon Sep 15, 2008 8:55 pm**

The last post should have read line -9, not line 9.

Posted: **Mon Sep 15, 2008 8:57 pm**

lamda1 is listed for both complex roots,

the second root should be lamda2.

Posted: **Mon Sep 15, 2008 10:42 pm**

In the first equation after Remark 2.3.13 k*exp(integral(h(t)dt)) should have a negative integral, k*exp(-integral(h(t)dt))

Posted: **Mon Sep 15, 2008 11:27 pm**

"example problems example" is redundant.

-Tom Banasiak

Posted: **Tue Sep 16, 2008 2:43 pm**

should the equation read

x(t)=c1e^lambda_1t = tc2e^lambda_2t

with the t added before c2?

-Ian Mills

Posted: **Tue Sep 16, 2008 3:45 pm**

In the second eqn on the page,

u_1('')x_1 + u_1(')x_1(') + u_2('')x_2x_2 <--- there are two x_2 s where there should only be one

Also in the last eqn on the page, the parenthesis around the integrals are inconsistent.

-Tom Banasiak

Posted: **Tue Sep 16, 2008 4:54 pm**

On the third equation in example 3.3.1, the right last term on the right hand side, exp(lambda*t) should be multiplied by a factor of six: 6*exp(lambda*t)

Posted: **Wed Sep 17, 2008 11:39 am**

On page 73 in example 3.3.1 the substitution is missing a 6 coefficient on the last term.

Posted: **Sun Sep 21, 2008 10:05 am**

In the series of simplifications, the last two are the same.

Posted: **Sun Sep 21, 2008 10:08 am**

"differential equation will yield and equation for mu that" the 'and' should be 'an'

Posted: **Sun Sep 21, 2008 10:10 am**

Theorem 2.3.8: "An nth dorder, linear, ... differential of the form..." should have 'equation' after differential.

Posted: **Sun Sep 21, 2008 10:12 am**

There is a pair of commas after 'Boit number,' should only be one.

Posted: **Sun Sep 21, 2008 10:16 am**

"Observe that that in Examples..." should be "observe that in ..."
also, there is no space between "and" and 1.6.4

Posted: **Sun Sep 21, 2008 11:17 pm**

The text reads: "A particle is a object that..."

I believe it should read: "A particle is an object that..."

The "a" before object was replaced with "an".

- Bristol

Posted: **Mon Sep 22, 2008 9:28 pm**

Beginning on line -4, the sentence that starts "This is intuitively appropriate in some cases, and is rigorously justified when the boit number..." is a sentence fragment. I believe the fragmentation is caused by the "when" just before "the boit number."

The sentence should conclude "is much less than one." and leaving the next sentence intact, or should be combined some other way with the next sentence beginning "When Bi << 1...".

Posted: **Tue Sep 23, 2008 2:26 pm**

The equation currently has c_2 (the coefficient to the sin(2t) term) equal to 2/13, but as shown in the c_2 value found three lines above it should be equal to 3/13. This also leads to line 19, in which the coefficient before the sin(2t) term should also be 3/13 for the same reason.

-Nate

Posted: **Tue Sep 23, 2008 3:44 pm**

In commented area of program, the differential equation written was actually the initial condition from the problem.

Should read:

x' = 75x(1-x)

not,

x' = 1/(1+exp(-10*(t-5)))

Posted: **Tue Sep 23, 2008 4:37 pm**

Preface: line 15 "important that proficiency" is "important than "
pg 43: 3 lines of Problems 1.2-4 (1.2) reads "approximation" (1.3,1.4) read "Approximate"
pg 60 Fist two equations at top The upper bound should be [x(t)] and in eqn 2 the lower bound should be [x(t)] and the (t)1/x dx shouldn't have the (t) term--which belongs in the bound
pg 63: L 1 "In contrast, to C^14, " no comma after contrast
P 63: Last bullet in summary section your first sentence is two dependent clauses (could remove the word "then")
P 67: Example 3.2.2 " and using Euler's formula the two" has no comma after formula
P 80: L (between h /ddot eqn and x_p(t) eqn) "While not necessary to simply find the solution" is a fragment, consider revising.
P 87: L (2 below first bulk of equations) "familiar substitution rule, in the " needs no comma after rule

Posted: **Tue Sep 23, 2008 7:34 pm**

"...with solving for the coefficients is WHEN the denominator..." (not "then")

Posted: **Tue Sep 23, 2008 8:20 pm**

in the second half of the equation, the second term of the denominator should be:

x1'(t)x2(t)

not: x1'(t)x1(t)

Posted: **Tue Sep 23, 2008 8:44 pm**

The text reads:

"A plot of v_o vs. [s] for various values of v_max and k_m are illustrated in Figures 2.1 and 2.2."

I believe it should read:

"Plots of v_o vs. [s] for various values of v_max and k_m are illustrated in Figures 2.1 and 2.2."

Bristol

Posted: **Tue Sep 23, 2008 9:41 pm**

the line reads "... and each case results in a solutions of a different type."

It should read "... and each case results in a solution of a different type."

"solutions" should be "solution".

-Mike Croteau

Posted: **Tue Sep 23, 2008 9:41 pm**

In the first equation on page 84, there are two plus signs in between -(1/2)tcos(2t) and (1/4)sin(2t). I think there is only supposed to be one.

In addition, in my post earlier regarding "pg. 84 line 3", I actually meant "pg. 85 line 3" regarding

"second order equation (equation 3.12) has been [replaced] with another one (equation 3.14)"

The word replaced appears to be missing.

Posted: **Tue Sep 23, 2008 9:43 pm**

[quote="mcroteau"]the line reads "... and each case results in a solutions of a different type."

It should read "... and each case results in a solution of a different type."

"solutions" should be "solution".

-Mike Croteau[/quote]

Sorry, the above should have said p. 73 line -11 not p. 74

Posted: **Tue Sep 23, 2008 10:00 pm**

The text reads:

"Since this equation is of the form of Equation 2.20, so the general solution is given by Equation 2.23 where h(t) =3/t and g(t) = sin(t)."

I believe it should read:

"This equation is of the form of Equation 2.20, so the general solution is given by Equation 2.23 where h(t) =3/t and g(t) = sin(t)."
(removal of "since" at the beginning of the sentence)

- or -

"Since this equation is of the form of Equation 2.20, the general solution is given by Equation 2.23 where h(t) =3/t and g(t) = sin(t)."
(removal of "so" after the comma)

Posted: **Tue Sep 23, 2008 11:17 pm**

The text reads:
"x(t) = c_1*x_1(t) + c_2*x_2(t)
= c_1 *cos(w_n*t) + c_2*sin(w_n*t)"

I believe it should read:

"x(t) = c_1*x_1(t) + c_2*x_2(t)
= c_1 *sin(w_n*t) + c_2*cos(w_n*t)"

On page 69, line -7/8 x_1(t) is defined as sin(w_n*t) and x_2(t) is defined as cos(w_n*t). The plugging in on page 70 just switches x_1(t) with x_2(t).

Posted: **Wed Sep 24, 2008 12:12 am**

The text reads:

"In the case in the mass-spring-damper..."

I believe it should read:

"In the case of the mass-spring-damper..."

The second "in" was replaced with "of".

Posted: **Wed Sep 24, 2008 12:35 am**

Problem 3.5 on page 89 is a second order, linear, ordinary, variable coefficient, inhomogeneous differential equation; these equations cannot be solved at this point in the course.

Posted: **Wed Sep 24, 2008 2:30 pm**

Incorporated through here.

Posted: **Sun Sep 28, 2008 6:52 pm**

The assumed particular solution has two dots above "x_p (t) = ", as if it were a 2nd derivative. Those shouldn't be there.

Posted: **Sun Sep 28, 2008 11:28 pm**

Sentence missing comma after Fcoswt.

Should read: Yet another interesting feature of this solution is apparent when one considers the relative phase between the forcing function, Fcoswt, and the response of the system.

Posted: **Mon Sep 29, 2008 12:22 am**

At the bottom instead of "...explain any significant phenomenal that you observe." I think it should be "...explain any significant phenomena that you observe." (changing the word phenomenal to phenomena)

Posted: **Mon Sep 29, 2008 5:15 pm**

"The term "single" refers to the fact they system has only one..."

"they" should be "the".

Posted: **Mon Sep 29, 2008 6:15 pm**

The first line of equation on this page should be for Xp, not Xp double dot.

Posted: **Mon Sep 29, 2008 6:17 pm**

Page 99, equation 4.8

The middle term should be (xdot_o / w_n) sin (w_n*t) as in 4.7 opposed to just the xdot / w_n

Posted: **Mon Sep 29, 2008 6:54 pm**

x'' + 2(zeta)(omega n)x

should be x'' + 2(zeta)(omega n)x'

should be first derivative of x, not x in the second term

Posted: **Mon Sep 29, 2008 10:39 pm**

Notes that Figure 4.3 shows solutions for natural frequencies 1,2,3,4,5. Figure 4.3 only shows 1,2, and 3.

Posted: **Tue Sep 30, 2008 1:12 pm**

"Now consider the problem of a coupled by a spring to a vibrating base. . ."

should be

"Now consider the problem of a [mass] coupled by a spring to a vibrating base. . ."

Posted: **Tue Sep 30, 2008 1:13 pm**

". . . that means the argument to the [since] function. . ."

[since] should be [sine]

Posted: **Tue Sep 30, 2008 1:27 pm**

". . . probably a idea, so it appears in Figure 4.20."

should be

". . . probably a [good] idea, so it appears in Figure 4.20."

Posted: **Tue Sep 30, 2008 1:30 pm**

vertical axis showing magnification of static deflection has axis label running into scale on axis

Posted: **Wed Oct 01, 2008 11:18 am**

Not sure if this qualifies as a typo, but the legends on figures 4.8 and 4.9 both have f(t) and x(t) overlapping, making it slightly difficult to read (possibly a software error).

Posted: **Thu Oct 02, 2008 6:01 pm**

The text reads:

"This is in contrast with multiple degree of freedom system, ..."

I believe it should read:

"This is in contrast with a multiple degree of freedom system, ..."

The word "a" was added to the sentence.

Posted: **Thu Oct 02, 2008 6:35 pm**

The text reads:

"... the case where omega << omega_n"

I believe it should read:

"... the case where omega << omega_n."

A period was added to the end of the sentence.

Posted: **Mon Oct 06, 2008 1:31 am**

Is Figure 4.16 correct? i graphed it and did not get the same result. also page 110 eq. 4.17 it should be xdot on the second term

Posted: **Tue Oct 07, 2008 8:57 am**

Also, in the homework solutions, Problem 4.3 is mislabeled as 4.2.

Posted: **Thu Oct 09, 2008 1:29 pm**

The solution becomes

x(t) = (the stuff already there) + c_1*cos(\omega_n*t)+c_2*sin(\omega_n*t)

if carried over from the equation before simplifying F/k.

Posted: **Sun Oct 12, 2008 8:12 pm**

I believe Equation 5.7 should also be multiplied by the term a_0.

Posted: **Sun Oct 12, 2008 8:18 pm**

The text reads,

"A very important properties of a series is its..."

I believe it should read,

"A very important property of a series is its..."

Properties was changed to property.

Posted: **Sun Oct 12, 2008 8:27 pm**

I believe the equation that is labeled, d^2x/dt^2 (t) = sum(n*(n-1)*a_n*t^(n-1))

should instead be: d^2x/dt^2 (t) = sum(n*(n-1)*a_n*t^(n-2)).

The last term, t^(n-1) was switched to t^(n-2).

Posted: **Sat Oct 18, 2008 6:44 pm**

"rather, one must compute resort to the chain rule as expressed in the equations above."

I believe "compute" should be taken out to fix the sentence grammatically.

Ashley Meklis

Posted: **Sun Oct 19, 2008 6:11 pm**

You have: "Hence, substituting for f(x(t),t) and f(x(t)+f(x(t),t)*dt,x+dt) into equation 13.15 gives"

I believe this should read: "Hence, substituting for f(x(t),t) and f(x(t)+f(x(t),t)*dt,t+dt) into equation 13.15 gives" where the x near the end of the quote was changed to a 't'.

Posted: **Sun Oct 19, 2008 8:14 pm**

In Equation 5.8, you have

k_d*(1-k_p) Should be k_d*(1-k_p)
------------ ==> ------------
k_p*sqrt(4kp-k_d^2) k_p*sqrt(4k_p-k_d^2)

The kp in the denominator should be changed to k_p (subscript).

Posted: **Mon Oct 20, 2008 11:43 am**

The v_3 equation is shown as: v_3 = (-(x(t)+v_2)^3 + sin((t+dt)(x(t)+v_2)))*dt, but it should be v_3 = (-(x(t)+2v_2-v_1)^3 + sin((t+dt)(x(t)+2v_2-v_1)))*dt

Posted: **Thu Oct 23, 2008 12:50 pm**

Sentence reads "As a matter of substance is is a matter of considering multivariable Taylor series expansions."

first is should be "it" so that the sentence reads "As a matter of substance it is a matter of considering multivariable Taylor series expansions."

Posted: **Sat Oct 25, 2008 1:07 pm**

The text reads,

"..., such as in Watt's steam engine governor [], it ..."

I believe it should read,

"..., such as in Watt's steam engine governor, it ..."

The "[]" was removed.

Posted: **Sat Oct 25, 2008 1:09 pm**

The text reads:

"This chapter provides an detailed introduction... "

I believe it should read:

"This chapter provides a detailed introduction... "

The word "an" was changed to "a".

Posted: **Sat Oct 25, 2008 1:29 pm**

The text reads:

"..., the particular solution is the same the proportional control case in..."

I believe it should read:

"..., the particular solution is the same as the proportional control case in..."

The word "as" was added to the second case.

Posted: **Sat Oct 25, 2008 1:57 pm**

The text reads:

"..., the relatively obvious thing to do is to starting including..."

I believe it should read:

"..., the relatively obvious thing to do is to start including..."

The word "starting" was changed to "start".

Posted: **Sat Oct 25, 2008 2:09 pm**

The text reads:

"A the two approximate solutions and... "

I believe it should read:

"The two approximate solutions and..."

The word "A" was removed.

Posted: **Sat Oct 25, 2008 2:14 pm**

The text reads:

"Because the form of the partial derivatives in equation 13.5, an example with the function f(x,t) that includes both x and t may be helpful."

I believe it should read:

"Because of the form of the partial derivatives in equation 13.5, an example with the function f(x,t) that includes both x and t may be helpful."

The word "of" was added.

Posted: **Sat Oct 25, 2008 2:19 pm**

The text reads:

"Similarly, df/dx and df/dt can depend on both x and t as well, so must be expanded similarly."

I believe it should read:

"Similarly, df/dx and df/dt can depend on both x and t as well, so they must be expanded similarly."

The word "they" was added.

Posted: **Sat Oct 25, 2008 2:28 pm**

The text reads:

" = [x(2.5) - 2x(1.5) + x(1.5)] / (0.5)^2 "

I believe it should read:

" = [x(2.5) - 2x(2.0) + x(1.5)] / (0.5)^2 "

The second term "2x(1.5)" was changed to "2x(2.0)". I believe there was a simplification error from the line above.

Posted: **Sat Oct 25, 2008 2:57 pm**

The last line of Equation 13.23 reads:

"xdot_n = f_1(x_1(t), x_2(t),..., x_n(t),t)."

I believe it should read:

"xdot_n = f_n(x_1(t), x_2(t),..., x_n(t),t)."

The first term to the right of the equal sign was changed from "f_1" to "f_n".

Posted: **Sat Oct 25, 2008 3:03 pm**

The text reads:

"Using this, the the Taylor series to second order..."

I believe it should read:

"Using this, the Taylor series to second order..."

The repeated word "the" was removed.

Posted: **Sun Oct 26, 2008 10:44 pm**

in the in the first and second equations on the page and equality for x2, I believe the "mgl" term is missing.

Posted: **Mon Oct 27, 2008 11:21 pm**

The text reads:

"..., and increasing kp increases the frequency of oscillation... ."

The "kp" in the above line is not italized and does not have the p as a subscript as it is done throughout the book.

Posted: **Mon Oct 27, 2008 11:24 pm**

The text reads:

"..., obviously, to expand f(x(t) + c_2*f(x(t),t)*delT, t + c_4*delT) in a Taylor series."

I believe it should read:

"..., obviously, to expand f(x(t) + c_3*f(x(t),t)*delT, t + c_4*delT) in a Taylor series."

The c_2 was changed to a c_3.

Posted: **Tue Oct 28, 2008 1:55 pm**

In the line that contains:

x = x +(v1 + 4*v2 + v3)/6.0

I think it should be:

x = x + (v1 + 4.0*v2 + v3)/6.0

because the 4 (changed to 4.0), should not be an INTEGER but rather a REAL number to not affect the final answer.

Posted: **Tue Oct 28, 2008 2:13 pm**

On page 548 line -4, the equation is f = 5*x. I'm not sure entirely how the double precision function command operates, but judging from what we learned about integers versus real numbers, this equation possibly should be f=5.0*x.

On page 549-551 line -4, the equation is f=-x**3+sin(t*x). Again, not being sure if this will make a difference with the function command, this equation should possibly be f=-x**3.0+sin(t*x), using a real number rather than an integer.

Posted: **Tue Oct 28, 2008 3:47 pm**

text currently reads:

"you may decided to write"

It should read:

"you may decide to write"

The tense of the verb was wrong.

-Nate

Posted: **Wed Oct 29, 2008 2:18 am**

Text reads
"As has already been stated, the dependence of f on x and t is specified, but the dependence of x on t."

This is grammatically incorrect. A possible correct sentence could read
"As has already been stated, the dependence of f on x and t is specified, but the dependence of x on t is not."

Posted: **Wed Oct 29, 2008 8:56 pm**

On page 442 in the paper copy of the textbook, "Using this, the the Taylor series " should be "Using this, the Taylor series". Repetitive "the"

Posted: **Thu Oct 30, 2008 9:51 pm**

"Not much thought...solution as x(t)=-cos t"
Should be "x(t)=cos t" (positive).

Posted: **Tue Nov 04, 2008 4:30 pm**

"If damping is light, however, the magnitude of the response may be large; however, unlike the undamped case, it does not grow unbounded."

This is more of a grammatical/style thing, but you may not want to use however twice in the same sentence and use "on the other hand" or "consequently" or another synonym that makes sense in the context.

-Nate

Posted: **Tue Nov 04, 2008 11:16 pm**

The line begins with "Example 9.2.7 finally..."

I believe the "finally" should be capitalized to follow the other formatting in the book, because it is the start of a sentence.

-Nate

Posted: **Wed Nov 05, 2008 12:17 pm**

Equation 13.18 should use an "approximately sign" instead of an = (as should the first two equations on that page)

Posted: **Wed Nov 05, 2008 12:21 pm**

Equation 13.21 should use an approximately sign rather than an = (as shoueld the 8th equation down)

Posted: **Mon Nov 10, 2008 9:42 pm**

The sentence says, "The string is assumed to elastic, which means it may have an internal tension, t, but it requires no moment to bend it."

There should be a "be" after the "to" so that the sentence reads "The string is assumed to be elastic, which means..."

Posted: **Mon Nov 10, 2008 9:43 pm**

Sentence says "Analogous to ordinary differential equation, in order..."

equation should be plural so that it reads "Analogous to ordinary differential equations..."

Posted: **Thu Nov 13, 2008 2:18 am**

the equation has both second order partials with respect to y. I believe one of them should be with respect to x.

Ashley Meklis

Posted: **Fri Nov 14, 2008 1:43 am**

"where the constant were combined into a_n and b_n"
"constant" should be plural.

Posted: **Fri Nov 14, 2008 1:54 am**

Here, alpha is defined as
alpha = sqrt(T/rho).

From Equation 12.4 (and elsewhere), I think it should be
alpha = sqrt(tau/rho),
as tau was the variable for tension, and T was the part of u(x,t) depending on time.

Posted: **Fri Nov 14, 2008 11:32 am**

I think "degree of 'alignment' the functions" should be "degree of 'alignment' of the functions."

Posted: **Fri Nov 14, 2008 11:34 am**

On the right hand side of the long equation:
the L should be the upper boundary of integration, not part of the integrand.

Posted: **Fri Nov 14, 2008 11:36 am**

in the 2nd line for u(x,0),
the infinity sign should be on top of the Sigma. Same thing a few lines later.

Posted: **Fri Nov 14, 2008 11:37 am**

in the first line for u(x,0)
the alpha should not be squared within the parenthesis.

Posted: **Sat Nov 15, 2008 6:37 pm**

"Proving following two force..."
should be
"Proving THE following two force..."

Below, in Problem 12.1, "the" is repeated.

Posted: **Sun Nov 16, 2008 12:19 pm**

The text reads:

"It is left as an exercise to show that if lambda =< 0 the boundary conditions can not be satisfied."

I believe it should read:

"It is left as an exercise to show that if lambda =< 0 the boundary conditions cannot be satisfied."

The words "can not" was changed to "cannot".

Posted: **Sun Nov 16, 2008 12:23 pm**

The text reads:

"... exhibit solutions that are oscillatory, which may the case for a second order equation."

I believe it should read:

"... exhibit solutions that are oscillatory, which may be the case for a second order equation."

The word "be" was added to the second version.

Posted: **Sun Nov 16, 2008 12:26 pm**

The left side of the equation in line 10 has the infinity sign coming after the sigma instead of on top of it. I believe the infinity sign should be on top of the sigma.

Posted: **Sun Nov 16, 2008 12:28 pm**

The text has a period after "u(x,b) = f(x)" (at line -3) but then the sentence continues with the word "is" and goes on to show the equation at the bottom of the page with period following that. I believe the period after the u(x,b) equation (line -3) should not be there.

Posted: **Sun Nov 16, 2008 3:21 pm**

In the equation at the bottom of the page,
I think the first term should be the 2nd derivative of u w.r.t. x, not y.

Posted: **Sun Nov 16, 2008 6:08 pm**

the sentence reads, "This explains why a plucked and struck sting..." it should read "This explains why a plucked and struck string..." (change 'sting' to 'string')

Posted: **Sun Nov 16, 2008 6:13 pm**

The x-label for the graph is labelefd as being time (t), however, it should be the position (x) along the string.

Posted: **Sun Nov 16, 2008 8:32 pm**

In equation 12.14, the lower bounds for x should be missing. It should be from zero to one.

Posted: **Mon Nov 17, 2008 1:41 am**

In the equation for b_o, the integration is divided by 2L. I believe it should only be divided by 1L.

Posted: **Mon Nov 17, 2008 1:42 pm**

Dr Goodwine,

In the equation beginning X(x) =...., the square root symbol contains both x and lambda in both terms, but it should only contain lambda.

Posted: **Tue Nov 18, 2008 12:26 am**

it says "Observe that f1(x) and f3(s) are well-aligned..." I believe it should be "Observe that f1(x) and f3(x) are well-aligned..." changing the (s) to an (x).

Also, 4 lines down from that it reads "...have a the same sign," it should read "...have the same sign," Remove the "a"

Posted: **Fri Nov 21, 2008 7:04 pm**

The equation at the top of the page should read:

T_2-T_1
=f(x) Over X - T_1
L

Posted: **Sat Nov 22, 2008 12:55 am**

the equation for Y(y) has "pi" instead of its symbol "\pi"

Posted: **Sat Nov 22, 2008 12:36 pm**

The square root in your general solution for X(x) covers the x in the sin and cos terms. It should just be the square root of lambda.

Posted: **Sat Nov 22, 2008 12:43 pm**

The infinite for the second calculation of u(x,0) should be on top of the sum not beside it.

Posted: **Sat Nov 22, 2008 12:46 pm**

After multiplying by sin the infinite should be on top of the sum, not beside it.

Posted: **Sat Nov 22, 2008 12:57 pm**

In the first computation of u(x,0) the alpha in the exponential term should not be squared.

Posted: **Sun Nov 23, 2008 10:54 am**

On page 395 you give the derivation of the heat equation, but never state what alpha is. It would probably be useful to do so.

Posted: **Sun Nov 23, 2008 10:13 pm**

in the paragraph at the bottom of the page it says, "In the case where the temperature profile has a nonzero, there will..." it should read, "In the case where the temperature profile is nonzero, there will..."

Posted: **Mon Nov 24, 2008 1:48 am**

In the first equation on the page, the quantity (T2-T1)/L should be multiplied by x.

Posted: **Mon Nov 24, 2008 6:09 am**

Text reads: Hence, the once dimensional heat conduction equation is give by

Text should read: Hence, the once dimensional heat conduction equation is given by

Posted: **Mon Nov 24, 2008 6:11 am**

in the equation of X(x), the second exponential value should be negative. i.e. exp(-sqrt(-lamda)x)

Posted: **Mon Nov 24, 2008 6:13 am**

The sine function should not be in this equation. sqrt(lamda)=n*pi

Posted: **Mon Nov 24, 2008 6:13 am**

in the equation it still reads the text pi as opposed to the symbol

Posted: **Mon Nov 24, 2008 8:07 pm**

"and the steady-state solution from Equation 12.23..."

12.23 isnt introduced until the next page, and it refers to Laplace's equation. I believe this line should say Equation 12.22, which is the general solution to the homogeneous equation.

Posted: **Tue Nov 25, 2008 3:42 pm**

The second term should be "-e^(-sqrt(-lambda)*x)", you don't have the negative sign before the square root.

Posted: **Tue Nov 25, 2008 3:42 pm**

In your first term (with the k1 constant), you have pi written out instead of the symbol.

Posted: **Sun Nov 30, 2008 5:00 pm**

The dot product reads

<sin(n*pi*x/L),sin(n*pi*x/L)>

The second term within the brackets should be sin(m*pi*x/L), meaning the n should be an m.

<cos(n*pi*x/L),cos(n*pi*x/L)>

The second term within the brackets should be cos(m*pi*x/L), meaning the n should be an m.

<sin(n*pi*x/L),cos(n*pi*x/L)>

The second term within the brackets should be cos(m*pi*x/L), meaning the n should be an m.

Posted: **Mon Dec 01, 2008 5:14 pm**

The equation reads

a_n = (lambda*a_n-2) / ( (n+k)^2 - gamma)

There should be a negative sign in front of the expression on the right, thus the equation should read,

a_n = - (lambda*a_n-2) / ( (n+k)^2 - gamma)

-- Arturo Garcia

Posted: **Mon Dec 01, 2008 5:46 pm**

From the Preface:

On line 13, "envelope" is spelled "envelop".

Posted: **Mon Dec 01, 2008 5:56 pm**

This might be a stretch but...

the text reads,

"or, substituting into Equation 12.31 gives"

Meaning, substituting equation 12.33 into 12.31, however, the equation is substituted in equation 12.32

Posted: **Mon Dec 01, 2008 10:14 pm**

the sentence reads

"We will assume that it is real since it is a coefficient in the ordinary differential equations for X(x) and Y(y) and we are seeking real solution to Equation 12.23."

There is an "a" clearly missing between seeking and real, so the sentence should read

"We will assume that it is real since it is a coefficient in the ordinary differential equations for X(x) and Y(y) and we are seeking a real solution to Equation 12.23."

Posted: **Mon Dec 01, 2008 11:11 pm**

In line 8 it reads, "While is appears that we only have one boundary...." it should read "While it appears that we only have one boundary..."

Posted: **Mon Dec 01, 2008 11:17 pm**

The sentence reads,

"We will proceed has we have done before..."

It should read

"We will proceed as we have done before..."

Thing changed: "has" to "as"

-- Arturo Garcia

Posted: **Tue Dec 02, 2008 5:27 pm**

I'm pretty sure one of the 2nd partial derivatives of u should be dx^2 not dy^2.

Posted: **Tue Dec 02, 2008 8:19 pm**

you have \partial utr \partial dt twice in the equation, the second one should be \partial uss \partial dt

Posted: **Tue Dec 02, 2008 8:43 pm**

"For a circular membrane, like a drum, because the boundary condition will hold at a fixed radius, as
opposed to fixed values of x and y, it will be much more convenient to solve it in polar coordinates."

the word "because" should be omitted

Posted: **Tue Dec 02, 2008 8:57 pm**

in the equation above this one, you define \gamma as m^2, in the next equation, you use c1 sin n\theta +c2 sin n\theta, this n should be an m

Posted: **Tue Dec 02, 2008 9:24 pm**

"... and taking the values of each at e we may..."

"at e" should most likely be eliminated

Posted: **Tue Dec 02, 2008 11:14 pm**

For the Fourier section, you have a recurring problem (3x). As per equation 12.16 on page 383, you should have brackets around the a_n and b_n terms (the summation applies to both). However, you forget to type in the brackets on later equations:

Page 388, Line 11
Page 388, Line -3
Page 389, Line 5

Posted: **Tue Dec 02, 2008 11:15 pm**

Under 13.7.1, "Consider, for example, the the function..." has repeating "the"s
A Few lines later, "delta(x) << 1 is a fixed" makes no sense. Missing a word, or unneeded "a"?

Posted: **Tue Dec 02, 2008 11:43 pm**

[quote="emille12"]In the equation for b_o, the integration is divided by 2L. I believe it should only be divided by 1L.[/quote]

On second thought, it's correct as it is in the book. Sorry! [Can we not delete old posts?]

Posted: **Wed Dec 03, 2008 12:05 am**

The y axis has the wrong label. It has the equation for b_n in the previous example rather than a_n in example 12.1.3.

Posted: **Wed Dec 03, 2008 12:07 am**

"A plot of the magnitude of the coefficient, b_n..."

b_n is really a_n.

Posted: **Wed Dec 03, 2008 12:48 am**

updated online text--pg 428 eqn: 12.33 -- under title of 2-D wave equation

(d^2)u / (dx)^2 + [the partial with respect to y] = . . .

both partials were with respect to y

Posted: **Wed Dec 03, 2008 1:05 am**

There is an x missing after (T2-T1)/L

Posted: **Wed Dec 03, 2008 1:24 am**

This says the rate of heat conduction higher where the gradient is steeper. Shouldn't there then be more heat conduction from the center to the right of figure 12.16 and less heat conduction from the left to the center? The gradient is steeper on the right side and flatter on the left side. The book says the opposite in section 12.3.1.

Posted: **Wed Dec 03, 2008 1:53 am**

Should Figure 12.15 be on the previous page since it corresponds to the example on that page instead of being in the head equation section?

Posted: **Wed Dec 03, 2008 3:41 am**

the infinity symbol should be above the summation symbol, not next to it

Posted: **Wed Dec 03, 2008 12:03 pm**

fyi, the page/line numbers i am referring to are in the updated pdf online, not the original course text

Posted: **Sat Dec 06, 2008 4:56 pm**

page 462:

The last sentence on the page says "if necessary, converting a system of higher order differential equations to s system of first order"

should change the s to an a.

Brent Neville

Posted: **Sat Dec 06, 2008 9:25 pm**

There is an addition sign missing between the 2x and (64...) terms in equation 14.9.
Also, should there be text between equation 14.8 and the text "...near xo=4. Substituting...?" Because it seemed a little discontinuous...

Posted: **Sun Dec 07, 2008 12:14 pm**

In remark 14.1.2, second sentence: "...numerical packages may give a slightly different answers" should not have the "a."

Posted: **Sun Dec 07, 2008 1:19 pm**

In "...higher order differential equations to s system of first..." the "s" should be an "a."

Posted: **Sun Dec 07, 2008 1:22 pm**

In "may simply be referred to a a or the differential equation," the first (unitalicized) "a" should be "as."

Posted: **Sun Dec 07, 2008 1:35 pm**

"the simply coordinate transformation" should be "...simple..."

Posted: **Sun Dec 07, 2008 5:38 pm**

The sentence reads "is actually fundamental feature of it." but it should read "is actually a fundamental feature of it." An "a" is missing.

Posted: **Mon Dec 08, 2008 3:48 am**

There needs to be a space between the name of the section "Geometry and Stability of Equilibrium Points in the Phase Plane" and the page number, "467".

Posted: **Sat Dec 13, 2008 1:06 pm**

In the second equation on page 116, there is an extra ) at the end.

Posted: **Sun Dec 14, 2008 12:07 am**

the caption is the same for both graphs on this page. the bottom graph should read, "delta x =0.0075 "

Posted: **Sun Dec 14, 2008 1:32 am**

the word "that" should be replaced with the word "than"

Posted: **Sun Dec 14, 2008 3:29 pm**

the ' + - ' should be ' - '.

Posted: **Mon Dec 15, 2008 11:07 pm**

the roots contain sqrt(zeta^1 - 1) instead of sqrt(zeta^2 - 1)

Bill Goodwine's Courses

Course Text

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