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Notes on Diffy Qs: Differential Equations for Engineers: Changes
December 27th 2016 edition, Version 5.0 (edition 5, 0th update):
Pagination changed due to the added sections, but numbering of existing
material has not changed (except for a couple of figures),
so the new version is completely compatible with
the old version.
- New sections: A short section 0.3 on more detailed classification of differential equations, and section 1.8 on exact equations,
an often requested topic.
- Expanded the discussion of slope fields in section 1.2 and added a
new figure. Thus all figures in chapter 1 are thus renumbered.
- Added a new figure for the salt brine example in 1.6
- New exercises in existing sections:
1.1.11, 1.1.12, 1.1.106, 1.2.9, 1.2.10, 1.2.11, 1.2.105, 1.3.107, 1.4.105,
1.6.7, 1.6.104, 2.1.105, 2.2.15, 2.2.16, 2.2.106, 2.2.107, 2.3.11, 2.3.106,
2.4.104, 3.1.105, 3.3.6, 3.5.4, 4.10.9
- Made the figure captions be italic and smaller, to differentiate from
surrounding text.
- Remove references to IODE. Unfortunately it is no longer kept up to date
and does not work with newest Matlab.
- Thanks to Sonmez Sahutoglu and his students for some of the suggestions above.
- Improve style and exposition in many places.
- Fix the errata.
October 20th 2014 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- New chapter, chapter 8 on nonlinear systems. The numbering
or pagination of existing chapters has not changed.
- Improve wording in d'Alembert section and mention the term "characteristic coordinates".
- Many fixes in style, grammar, and exposition.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
December 18th 2013 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Add a note on the existence of Laplace transform for exponential order
functions. Also add one for using this property to compute the derivative.
Thanks to Dusty Grundmeier for the suggestion.
- Many fixes in style and grammar.
- Fix the errata.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
April 29th 2013 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Add some exercises that require a little more thought: 0.2.12, 1.2.7, 1.2.8, 4.9.11.
- The coffee cooling example had unrealistic temperatures. Plus Bob would
burn his tounge at 70 degrees. So change to more realistic temperatures by
actually measuring a cup. Add exercise 1.3.106 to deal with what kind of errors we get when the measurement is imprecise.
- Mention partial fractions after example for getting Taylor series for rational functions.
- More correct note on visualization of the Dirichlet problem at end of 4.9.
- Box the formula for Laplacian in polar coordinates.
- Minor grammar and style fixes.
- Fix the errata.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
December 17th 2012 edition:
The added sections change page numbers beyond the new section 4.10 of course.
No other numbering changes occurred.
- New section 6.4 on the Dirac delta function. This was a frequently
requested addition. This adds 13 exercises, 5 of which have solutions in the
back.
- New section 7.3 on the Frobenius method. This adds 12 new exercises, 4 of which have solutions in the back.
- New section 4.10 on the Dirichlet problem in the circle and the Poisson kernel. This adds 12 new exercises, 4 of which have solutions in the back.
- Refer to Robin condition by name in chapter 5 and add it to index.
- In section 7.2 several equations were numbered (by mistake),
but never referred to, so stop numbering them.
- Fix references to Edwards-Penney in chapter 7.
- Footnote counter is reset per page as is usual,
so that we don't go into weird footnote symbols. In the web version use
numbers for footnotes since symbols are worse in that case.
- A few minor style improvements.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
October 1st 2012 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Exercise 4.3.103 changed slightly. Added a factor of n (actually
n+1) to the denominator to avoid questions of convergence and differentiability.
- Improve the paragraph about decay rates of the coefficients. It implied
something that was not correct (marked as errata). Of course what we said only holds for the
types of functions we've been defining. Also note what decay rates guarantee
continuity and derivatives.
- Improve slightly the exposition on integrating factor.
- Lots of minor improvements in style and fixes in grammar, for example there were a lot of missing question marks
in exercises.
- Add exercises 1.1.105, 1.3.105
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
July 2nd 2012 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Fix all the errata.
- Add exercises 1.2.6, 1.2.104, 2.5.9, 2.5.105, 7.1.11, 7.1.105
- Replace duplicate exercise 6.3.104 with a different exercise (it was the same as 6.3.103).
- Add a tiny bit more detail in several places.
- Minor style and grammar fixes and improvements throughout.
- lulu.com have raised their prices a tiny bit so the paperback version is now 15 cents more expensive.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
April 8th 2012 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Fix all the errata.
- Add exercises 4.3.10 and 4.3.105
- Minor style, grammar, and spelling fixes.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
December 25th 2011 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
Thanks to Sonmez Sahutoglu for the following suggestions
- Mention the interval where everything is defined in theorems 2.1.2 and
2.3.2.
- In Theorem 2.1.3, no need to mention f.
- Rewrite Theorem 2.5.1 a little to use "complementary solution"
for $y_c$ and only say it is the general solution to the homogeneous
equation in parenthesis.
- In section 3.5 mention that we only consider diagonalizable matrices.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
October 24th 2011 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Fix all the errata.
- Add exercise 2.2.105.
- Minor style and grammar fixes.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
June 17th 2011 edition:
Pagination has changed somewhat in this version. Numbering of chapters,
sections, and exercises did not change. Only one subsection in Laplace
transforms chapter was added. The following changes were made:
- Add 146 exercises with solutions in the back
(and 1 without) bringing the total
to 485 exercises.
- Add subsection on transfer functions to section 6.2.
- Fix all the errata.
- Improve definition of linear independence for vector valued functions and
add an example. Thanks to Paul Pearson for pointing this out.
- In section 1.5, improve the table for rules of thumb for the substitution
by adding "v=". Thanks to Jared Teague for the suggestion.
- Lots of minor cosmetic and grammatic fixes.
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
December 9th 2010 edition:
Pagination and numbering was kept as constant as
possible. The following changes were made:
- Change the matrix in example of 2x2 defective matrix on page 125,
so that it is not the same as the matrix from exercise 3.7.2 (to prevent
"solution by looking ahead in the book")
- Fix all the errata in example 6.2.2, and make the step function
go from 1 to 5 (which actually matches the graph on page 240).
- Add Exercises 1.7.7 (Runge-Kutta), 3.8.10, 3.8.11, 6.1.13, 6.1.14, 6.1.15,
6.2.10, 6.2.11, 6.3.8, 6.3.9
- Fix many grammar mistakes and typos (thanks to Gladys Cruz, Jonathan Gomez, Janelle Louie, Navid Froutan, Grace Victorine)
- On bottom of page 233, when expanding in partial fractions,
the $A(s^2-1)+s(Bs+C)$ should be
$A(s^2+1)+s(Bs+C)$. The result is correct.
- On page 256, the explicit expression for $a_5$ was wrong,
it should be $a_5 = \frac{a_1}{(5)(4)(3)(2)}$ (thanks Gladys Cruz)
- On page 260, the $a_2$ expression on the top was missing a
negative sign (thanks Jonathan Gomez)
You can get the
LaTeX source
or
PDF
of this old version
if your really want to, but I suggest just using the newest version.
November 15th 2010 edition:
Except for a new chapter, pagination and numbering is kept as constant as
possible. The following changes were made:
- New chapter, chapter 7 on Power series methods. The numbering
or pagination of existing chapters has not changed. This is a short chapter
for 2-3 lectures.
- In Figure 3.3, remove the lines spanned by the eigenvectors to make the
eigenvectors themselves clear. The lines are there in Figure 3.4.
- After theorem 3.4.1, and in example 3.4.4, give the corresponding
fundamental matrix solution.
- Page 100, fix computing the determinant of the matrix, it is
$(2-\lambda)((2-\lambda)^2-1)$. There was an extra power of 2.
- In Example 3.4.4, the general solution was missing the arbitrary
constants $c_1$, $c_2$, and $c_3$. Furthermore,
the general solution had the wrong eigenvector for $\lambda=2$.
- On top of page 108 when showing the spiral sink, we meant to take real and
imaginary parts of $\vec{v}e^{(-1-2i)t}$. The displayed equations were
correct though.
- On bottom of page 121, when the method was being generalized, the equations
had 3 in them where there should be $\lambda$.
- Links to Wikipedia (at least when viewing online) from the short bio footnotes.
- Lots of minor clarifications, grammar fixes, and improvements.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
November 1st 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Add exercises 3.2.11, 3.4.11.
- Add theorem 3.4.2 summarizing the complex eigenvalues situation.
- Add references to Boyce-DiPrima in the section notes as this
is a commonly used textbook.
- Add examples for when matrices do not commute and
when AB=0 does not imply that A=0 or B=0 for matrices.
- Add vertical bar to augmented matrices.
- Page 58, bottom, fix end of Example 2.3.1. (Thanks to Joanne L. Shin)
- Fix labeling of the masses in figure on on page 84.
- On page 85, fix the wording in Example 3.1.2.
- Lots of minor fixes and clarifications.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
October 13th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Fix example on page 38 when h=1.6
- In exercise 1.4.9 part a) we simply ask for the concentration as a function
of time (not requiring conversion of time to seconds)
- On top of page 97, $\vec{x}_p$ was missing the arrow indicating a vector.
- Theorem 3.7.1 had a typo where instead of "then" there was an "and".
- On top of page 120, it said the "number of eigenvalues" was equal to the
number of free variables in $A\vec{v} = \lambda \vec{v}$, when of course it is
the "number of eigenvectors".
- When discussing variation of parameters, we say that the equations given
work for any second order Ly=f(x). Of course, here we must have L of
the form $y''+p(x)y'+q(x)y$.
- Define "linear combination" more explicitly.
- Mention Boyce-DiPrima in the introduction and in further reading.
- Improve the cart figure in beginning of chapter 3.
- Lots of minor fixes and clarifications.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
October 3rd 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Add small drawing of the water tank to example 1.4.2
- Add remark about solutions to linear equations (remark 1.4.1)
- Fix example 1.1.5 (missing factor of in the solution 2).
Thanks to Wing Yip Ho.
- Clarified the stuff about numerical instability (rewrote that paragraph)
in 1.7 and mention the word "stiff"
- Add Exercise 1.7.6
- Many minor clarifications and cosmetic/grammar fixes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
July 16th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Use the microtype latex package for generally nicer line breaks.
- Make all tables prettier.
- Remove some useless significant digits from the tables in the
Euler's method section.
- Improve linking.
- Remove "Dirichlet problems" from the Steady state temperature
section title as it didn't fit properly.
- Many minor clarifications and cosmetic/grammar fixes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
July 1st 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Using new texlive, which outputs PDF 1.5, which is significantly smaller.
- Underline links, the underlines should not appear in printed output.
- Some minor clarifications and cosmetic/grammar fixes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
May 18th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Exercise 4.6.5 was ridiculous in light of exercise 4.6.4. I replaced
the exercise with a somewhat more difficult one.
- On page 113, when writing the solution in terms of amplitudes and phase shifts, $\alpha_1$ and $\alpha_2$ were swapped.
- Exercise 118 on page 118 wrongly asked for
$m_1$ instead of $m_2$ which is the unknown.
- On page 198 fix typo in the derivation of d'Alembert formula.
Thanks to Shawn White.
- Add exercise 6.2.9
- Many minor grammar/English fixes and small clarifications.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
April 28th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- On page 193 fix the
derivation of the solution for w and z.
- On page 173 the formula near the bottom should have a -4. That is, f(t) is
$\frac{1}{2} + \sum_{{n=1\\n~\text{odd}}}^\infty
\frac{-4}{\pi^2 n^2} \, \cos (n \pi t)$. Thanks to Sean Robinson.
- On page 206
the formula for $u_n(x,0)$ should read
$u_n(x,0) = X_n(x)Y_n(0) = \sin \left( \frac{n \pi}{w} x \right)$ (note the denominator).
Thanks to Sean Robinson.
- Added note to heat equation section about the ``smoothing out'' effect of
the solutions. For whatever reason, this note wasn't in there even though it
is referred to in the wave equation section.
- Add exercises 4.4.11, 4.6.10
- Fix the pagination in chapter 4 to 1) make it similar to what it always has been, and 2) to avoid bad page breaks
- Add more acknowledgments.
- Several small clarifications, and other minor fixes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
April 13th 2010 edition:
Pagination (pagination changed in chapter 4 and is somewhat odd, that will be fixed in next version) and numbering is kept as constant as possible. The following
changes were made:
- Many clarifications as to arguments of sin/cos/sinh/cosh, that is, use more parentheses. In fact,
any argument that is not a single symbol is now parenthesized, and some even if they are single symbol.
It is not as pretty, but it's a lot more readable.
- On bottom of page 173, there was the constant term of 1/4 missing in the
Fourier series for the solution x.
- On page 237, Heaviside function definition was off it should be 1 for all t ≥ 0.
- On page 239 near middle of page, there was an
$e^{-2s}$
that should be
$e^{-3s}$
though the inverse transform was then done correctly. Thanks to Sean Raleigh and Jessica Robinson for
spotting this one.
- On page 240, when applying the Laplace transform to solve an integral equation, there
was a missing equals sign and x(t) was mistakenly called f(t)
- On page 241, exercises 6.2.3, 6.2.4, and 6.2.5 were trivial as given (x=0
is the solution). What I means was to make the initial conditions arbitrary,
i.e. $x(0)=a$ and $x'(0)=b$. Thanks to Sean Raleigh and Jessica Robinson for
spotting this one.
- On page 242, exercise 6.3.2, change all $\omega_0$ to $\omega$.
Thanks to Sean Raleigh and Jessica Robinson for
spotting this one.
- Add exercise 6.2.8.
- Mark example 6.2.4 as an example.
- Minor other clarifications and cosmetic improvements in places.
- Fixed many minor typos and mistakes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
April 6th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Exercise 4.1.4 on page 145 had forgotten applications of chain rule in computations. The final result is not changed. Thanks to Michael Angelini for pointing this out.
- The eigenfunctions are cos kt (not sin kt) on top of page 146.
- Clarify and add more information to exercise 2.6.5
- Page 153 was a little confusing, consistently use 1 as the constant
eigenfunction and frame the formula for $a_0$ at the bottom of the
page.
- Also on page 153, define the inner product without scaling by
$\frac{1}{\pi}$ for consistency.
- Clarify the definition at endpoints for periodic functions.
- On page 157-158 stay with the square wave example to talk about the
values at discontinuities to avoid jumping around.
- On page 163 top, the condition about existence of one sided limits at
all the endpoints is now correct.
- Be less cavalier with equality in the Fourier series section. Remove
a couple of unnecessary "f(t)=" before a series, before we have talked
about convergence.
- Clarify exercises 4.3.6 and 4.3.7
- Add exercises 4.3.8, 4.3.9, and 6.1.12
- On page 168 of course it is the cosine terms disappear for an odd
extension, not the sine terms.
- On page 170 on the top, there was a plus sign missing in the series
after
$\frac{a_0}{2}$.
- On page 212 we mistakenly said that eigenvalues are lambdas with no nontrivial solutions (thanks to Sean Raleigh).
- On page 217, the integral that is the inner product of the sine with itself should not have f(t) in it, the one that equals
$\frac{\pi}{4}$
(thanks again to Sean Raleigh).
- On page 222, Newton's second came out all jumbled up. Of course it is "force equals mass times acceleration"
(and again: Sean Raleigh).
- Add hint to exercise 5.1.1 as there is a subtle issue.
- Minor other clarifications and cosmetic improvements in places.
- Fixed many minor typos and mistakes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
March 7th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Exercise 2.4.5 was not possible. Replace 0.8 with 1.1 and 0.39 with 0.8.
Also add a note that you always get two possibilities for the mass
in the formulas.
- Fix the formula for the complex roots on page 54, that is the discriminant should be negated under the square root sign if we take out an i.
- Fix theorem 4.1.2. There was an unneeded hypothesis about p and q which do not appear in the theorem.
- On page 60, fix the solution corresponding to complex roots.
- On page 60, fix the factorization of the characteristic equation in
example 2.3.5.
- Also on page 60, relabel the arbitrary constants to be more consistent
and make them all upper case.
- IODE Project III has changed (there was always some confusion about what was Project III)
and now fits more after section 2.3, rather than before 2.6.
- On page 75 on top, there was a typo in the expression for $u_1'$, the x should be an argument to the cosine.
- In example 2.6.1, when stating the parameters, it should say m=0.5
(the example was computed correctly, so no other changes).
- Add exercise 2.4.6.
- Add hint to exercise 2.4.1.
- Theorem 2.5.1 holds for any order, so simply drop the words "second order"
- In exercise 2.5.1, also explicitly state that the initial conditions must
be satisfied.
- Remove unneeded reference to [EP] on page 73.
- Put a box around the formula for the practical resonance frequency on page
81.
- Minor clarifications in places.
- Add a few more words to the index.
- Fixed other minor typos and mistakes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
February 17th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Fix solution to example 1.3.4. The solution should end up being
$\frac{6}{x^2+2C}$.
Thanks to Leonardo Gomes for noticing.
- Fix the computation of the constants in Example 2.4.1. Also fix the
discussion right after the example, and fix Figure 2.2.
- Fix examples on page 72 and 73.
- Fix example 6.1.7 (page 234), the inverse transform was off by an extra 1/2.
- Fix example 6.3.3 (page 243), the convolution was not computed correctly.
- Figure on page 37 (bottom of page) should have x=5 and x=0. Thanks to Jeff Winegar for pointing it out.
- Add Theorem 2.1.3 to highlight the fact that the superposition of two linearly independent solutions is the general solution to the homogeneous equation.
- Add Theorem 2.5.1, to give the solution to highlight how nonhomogeneous
equations are solved.
- Add k to the examples on top of page 49 to make them less trivial.
- Exercise 2.3.4 was hard to read correctly, reformulate.
- Due to minor changes in the text, Figure 4.12 moved up a page resulting in
a much more logical pagination.
- Many small clarifications.
- Many minor typo and grammar fixes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
February 8th 2010 edition:
Pagination and numbering is kept as constant as possible. The following
changes were made:
- Fix error in example on page 34. The integrating factor is off by
a constant. Instead rework the example to be done by substituting into
formula (1.4). Thanks to Sean Raleigh for noticing.
- Integrate in a different way in Example 1.1.3 to get rid of an extra k.
Also fix up the constant in front. Finally rename C' to D and add assumption
that k > 0.
- Fix the derivation in example 1.3.3, there was a negative sign mistake in
the middle, though it made no difference in the solution. Also consistently
keep time in minutes for the example.
- Fix example 1.4.1, there was a missing minus sign in the solution. Thanks
to Ian Simon for pointing it out.
- Fix example on page 103. The expression for $x_1$ in the middle of
the page should have $e^t\cos(t) -i e^t\sin(t)$. The rest of
the example also needed to have the signs fixed. Thanks to Sean Raleigh
for noticing.
- Exercise 1.1.7 had a typo making the solution very hard (without a computer
algebra system). The right hand side should be a
$\frac{1}{y+1}$
not
$\frac{1}{y^2+1}$.
- Exercise 1.3.7 was impossible (typo in the equation) replace with a
different exercise.
- Add forgotten initial condition to Exercise 0.2.9 ($x'(0)=0$).
- Exercise 1.5.6 was way too difficult. Replace with an easier exercise.
- Add "then find x(10)" to exercise 1.1.1.
- Fix caption on Figure 1.
- Fix captions on Figure 1.7-1.10 (replace -0.1 by 0.1), replace Figures
1.6-1.10 with new versions which have the correct vertical scale.
- In example 1.2.2, use
$\operatorname{sgn}(x)x^2$
for an example solution.
- Add exercises 0.2.11, 1.1.9-10, 1.2.5, 1.3.8-12, 1.4.11-12, 1.7.5
- Exercise 1.5.1 was already worked out as an example. Replace.
- Regenerate some of the figures for better looking output using better
parameters and newer version of genius.
- Clarify exposition somewhat in many places, and clarify some exercises.
- Fix some other minor grammar mistakes or simply awkward English in places.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
January 10th 2010 edition:
Pagination, numbering, everything is the same except for very minor
edits as follows.
- Fixed examples on the first page of section 2.1. The k
in the equations should be k^{2}. Thanks to Thomas Wicklund
for noticing.
- Got rid of the version with the EP sections in the titles. So there is now
only one PDF version of the notes. If you like you can make any version you
want from the latex source.
- Restructured the initial "notes about these notes" section a little.
- Fix some other minor grammar mistakes or simply awkward English in places.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
November 25th 2009 edition:
Pagination, numbering, everything is the same except for very minor
edits as follows.
- Exercise 2.1.8 was missing a minus sign (thanks to Eliot Brenner for spotting this). It should look like:
Exercise 2.1.8: Suppose
$y_1$ is a solution to $y''+p(x)y'+q(x)y=0$.
Show that
$y_2(x)=y_1(x)\int \frac{e^{-\int p(x)dx}}{(y_1(x))^2} dx$
is also a solution.
- Minor English, grammar, spelling, and punctuation fixes all over.
- Minor clarifications, and some cosmetic changes.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
October 23rd 2009 edition:
Pagination, numbering, everything is the same except for very minor
edits as follows.
- Exercise 4.6.4 on page 188 is fixed.
While theoretically the exercise is possible, it
is not what was intended. The last side condition should be
u(x,0) = 3 cos x + cos 3x.
- Minor English, grammar, spelling, and punctuation fixes all over.
- The license in the book now explicitly specifies that printing course packs
is permitted. Plus other minor updates in the introductory section.
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
September 29th 2009 edition:
Pagination, numbering, everything is the same except for very minor
edits as follows.
- Example 1.4.2 is fixed. There was an error with integration and there
is a factor of 2 missing in the 5th line of the computation on page 30.
If you do not wish to get the whole new PDF file,
here is an updated page 29 and 30.
Thanks to Thomas Wicklund for spotting this error.
- Minor English, grammar, and punctuation fixes all over chapter 1.
- Website added to notes (chapter 0).
You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.
May 12th 2009 edition:
First version. You can get the LaTeX source
of this old version
if your really want to, but I suggest just using the newest version.