[Go to the Notes on Diffy Qs home page]

**May 15th 2024 edition, Version 6.7 (edition 6, 7th update):**

The point of this revision is to fix errata, and also to improve wording, clarity, and consistency in many places. It may seem to have lots of changes but they are all minor general polish kind of things I found while teaching. There is essentially no new material, except for a few short paragraphs in the Laplace transform (periodic functions, and expanded explanation for transfer functions), and a couple of places throughout where explanations were expanded or added. Most changes are in chapters 1, 2, 6, and 7, because those were the ones I used in class. Several exercises needed to be changed to fix errata.

- After Example 0.2.1, add a short note about how the analytic solution helped us find the parameters and how this is useful in real problems.
- On page 23, emphasize a bit more that if we applied the inverse function theorem, we are integrating with respect to y.
- On page 33 in Example 1.3.1, explicitly separate variables before integrating.
- In the Example in subsection 1.3.2, use \(D\) consistently for the arbitrary constant.
- Be more precise to say that "singular solution" is the solution that comes about because of the division by zero.
- In Example 1.4.2 at the very end, explicitly say in English that there are 11.86 kg of salt in the tank when full. Also in the remark about concentration afterwards, write 0.1667 to approximate 1/6 to use the same number of decimal places as the concentration.
- On page 58, add a displayed equations showing what the error is for clarity.
- In 1.9, mention again the meaning \(u_x\) and \(u_t\) notation for partial derivatives.
- Add "characteristic curve" to the index for 1.9
- On page 82 in the computation on top of the page add underbraces to make it clearer what's happening.
- Resonance is a terrible example for a doubled root on page 86, replace it with a reference to a critically damped mass-spring system. Add a remark about why two very close roots are bad (numerical instability).
- At the end of subsection 2.3.1, the reason why we want linear independence was not explicitly stated so add Theorem 2.3.3 to have a parallel with the three theorems from section 2.1. It is mentioned in passing in the next subsection, but this makes more sense.
- On page 96, explicitly say that \(t\) is in seconds and \(x\) is in meters since we give the units of all the other variables, also we forgot to say that current is measured in amperes.
- On page 105, add an explicit remark about trying a generic polynomial even if some of the powers do not really appear on the right hand side.
- On pages 106 and 107, some of the particular solutions were just $y$, but that's not totally consistent with the rest of the section.
- On page 109, the computation of the example for variation parameters, rewrite the \(u_1'\) in a different way to make calculus easier and use a different, somewhat simpler, antiderivative of the secant (the one it seems the calculus books I have use). Also be more consistent with not using parentheses after trig functions if there is only one letter, it does seem a bit more readable this way.
**Exercise 3.8.15 part b) was changed to fix erratum**, the matrix given was a typo that made the problem unnecessarily difficult and the solution in the back was what was intended.**Exercise 4.3.106 part f) was changed to fix erratum**, we really wanted to ask about F(-9), so that we also ask for some point not in the original range that's not the discontinuity. The answer in the back was in fact giving F(-9) as intended.- On page 286, do not use \(F(t)\) for the forcing function as capital \(F\) is used later in the d'Alembert solution so avoid confusion. Also explicitly use the words "forcing function" when we first talk about it to make the exercises easier to figure out.
**Exercises 5.3.6 and 5.3.101 were changed to fix errata**: As given, 1) the forcing functions would cause resonance and we didn't really cover that and 2) they were given in x rather than t by mistake. Both were changed to be on the interval \([-\pi,\pi]\) which avoids the resonance and makes computations easier anyhow.- In Theorem 6.1.2, mention the interval \([0,\infty)\) just in case someone wonders what happened to the \(\frac{1}{t}\) counterexample.
- In Example 6.1.7, add an intermediate step in computing the unshifted inverse transform.
- In 6.2.1, we offhand make a remark about piecewise smooth functions, but if the functions have discontinuities, then one would have to write a little bit more (the same procedure works, but does not get the same formula as we'd be missing the delta functions in the derivative). So just say "continuous" here and then the same formula will work.
- Replace Example 6.2.5 by a more intelligent one that also very simple, somewhat easier to understand, and not as trivially solved without Laplace (still easily though).
- Add short note about how one can compute \(H(s)\) without knowing the equations, just knowing \(X(s)\) for a given \(F(s) .\)
- Add an extremely short subsection 6.2.6 on Laplace transform of periodic functions, which seem to pop up in these courses, and it makes sense in the same section as where we are looking at piecewise functions.
- Add Exercises 6.2.13 and 6.2.104 on transfer functions from input and output (this is similar to the exercise on the WeBWorK).
- Add Exercise 6.2.14 for a periodic function.
- Clean up wording for exerecises 6.2.12 and 6.2.103.
- On page 309 add distributive law to the list of properties of convolution.
- In Example 6.3.4 mention that \(\frac{1}{s^2+\omega_0^2}\) is the transfer function (and flip the order to be consistent with the text before). Then after the example mention how to get the general formula from the transfer function, thus relating transfer functions and convolution.
- Use "rectangular pulse" consistently on 6.4 and put it in the index.
- Add a short note at the end of 6.4.1 to note that the Laplace of the given pulse function goes to 1.
- On page 314, when writing the convolution with delta, be more consistent.
- On page 316, after Example 6.4.2, write convolutions all in the same order for consistency, use \(h\) for the impulse response for consistency and to not jump back and forth with \(x\) and \(y ,\) in general make the wording clearer about how the computation works in general, and finally mention that the impulse response is just the inverse Laplace transform of the transfer function.
- On page 330, when writing out the Taylor series, mention that it is a power series to drive that point home and also expand a few terms as it is sometimes not totally clear what the \(k=0\) term means here.
- Expand the first few terms of the differentiated series on page 331 to make it clearer what it looks like.
- In sections 7.1, 7.2, 7.3, add 0.5 lecture to the estimates in the note for instructors, they were too optimistic. I just did them each in 2 lectures and I felt it was about the same pace as chapters 1 and 2 where I was hitting my estimates spot on.
- In sections 7.2 and 7.3, name the coefficients \(P(x),\) \(Q(x),\) \(R(x)\) since lower case letters are used for other things, most confusingly little \(r.\)
- In 7.3, say a little bit more explicitly in the beginning that we're just trying to find a solution on one side of the singular point. Also mention the thing about what the singular point does to the initial value in the first example.
- In 7.3 when describing the method, mention very quickly what to do if \(x_0\) is not zero.
**Exercise 8.4.101 part c was changed to fix erratum**: The system is changed to \(x'=xy^2,y'=x+x^2 ,\) and then the solution in the back (changing \(y\) to \(y^2\) of course) works.- Fix errata.

**October 26th 2023 edition, Version 6.6 (edition 6, 6th update):**

**Change Exercise 3.1.106**to fix an erratum. Rate of flow from tank 2 back into tank 1 should be \(r-s ,\) otherwise volumes do not stay constant and the problem doesn't quite all make sense.- Add a couple of minor clarifications to the end of Example 1.9.3.
- Slightly tighter spacing in 3.1 leads to a slight change (for the better I think) in pagination in the section.
- In the HTML version, add light horizontal rules to visually separate out figures and tables.
- A few minor clarifications and improvements in English and style.
- Fix errata.

**July 18th 2023 edition, Version 6.5 (edition 6, 5th update):**

- In 0.3, add the nonlinear pendulum equation as an example nonlinear equation.
- Also in 0.3, say a bit about difficulty of nonlinear equations, to give some motivation for why we study linear equations
- Also in 0.3, give the general form for an autonomous first order ode to drive the point home.
- Move Example 1.1.1 up by a paragraph as it makes more sense that way.
- In Example 1.4.2, explain where the \(2t\) comes up for figuring out when the tank is full, and emphasize that the concentration is when we divide by the volume, by the 100.
- In Example 1.5.2, note what \(F(v)\) is for clarity.
- Also in Example 1.5.2, note just before that since we have to avoid \(x=0\) we might as well assume \(x > 0\) or \(x < 0 . \) Use \(x > 0\) in the example for simplicity and therefore get rid of most of the absolute value signs.
- In 1.6, add a short note about plotting \(y=f(x)\) to figure out the signs.
- In 1.7, tighten the language when describing first vs. second order method, leading to slightly nicer page breaks as a side benefit.
- In Example 1.8.7, in the final solution for \(F(x,y)\) we better put absolute values around \(y .\) In a sense it is not wrong but we would require the complex logarithm otherwise.
- When talking about "linear operator" after Theorem 2.1.1, mention that it seems almost as if we are multiplying by it.
- In Exercise 2.1.6, explicitly mention the name of the equation and say "see above" as the explanatory note is on the previous page.
- A few minor clarifications.
- Several improvements in wording and fixes for grammar and style.
- Fix errata.

**November 20th 2022 edition, Version 6.4 (edition 6, 4th update):**

- Reformulate end of Example 1.8.5 so that it's clear that at the end we are talking about the corresponding \(M \, dx + N \, dy = 0\) equation. Also note that \(x=-1\) itself is a solution to that.
**Change Exercise 6.4.5 to fix erratum**, in particular, the impulse response given is changed to \(t e^{-t} . \)**Change Exercise 6.4.103 to fix erratum**, in particular, the impulse response given is changed to \(e^t \sin(t) \) and the equation whose solution is sought is changed to \(Lx = e^t .\)- Add Remark 6.4.2 after Example 6.4.2, which explains what's going on with the \(x'(0)=0\) by noting that \(x'\) has a jump at \(t=0\) to produce the delta function.
- Add footnote when defining the integral of the delta function that we are always taking the endpoints.
- Make discussion of the radius of convergence based on the ratio test a bit more readable.
- In 8.5 when talking about the Lorentz system attractor, be clearer by explicitly saying that the solution tends to the attractor.
- Due to changes in LaTeX (namely the newpx fonts), pagination has changed very slightly. In the new version the theoremfont option seems to be buggy so go back to regular punctuation in theorems and exercises, one can't even tell.
- In the HTML version, tables are now numbered properly.
- Other minor improvements in wording and style.
- Fix errata.

**May 6th 2022 edition, Version 6.3 (edition 6, 3rd update):**

**Change initial condition in Exercise 1.1.103 to fix an erratum.**The previous initial condition meant looking at a point where the differential equation is undefined. While one can solve the problem continuously, and the formal process just works, it's best to avoid it. The initial condition is changed to \(x(0)=\frac{\pi}{4}\) where the equation is well behaved.- In the process of fixing an erratum from Martin Irungu (a sign error in the "proof" of Theorem 6.1.2; the final answer is off by a sign), implement Martin's suggestion to write the whole thing in terms of the positive \(s-c\) rather than the negative \(c-s,\) which clearly leads to more typos.
- For some reason a new version of LaTeX is now making some spacing slightly different (being a little bit more generous with space under figures for example), so the pagination is slightly different in places. I went through and tried to get rid of bad page breaks by tightening up the language where needed (a good thing to do anyhow), so it is only in a couple of places where page breaks changed.
- Some minor wording improvements.
- Fix errata.

**June 9th 2021 edition, Version 6.2 (edition 6, 2nd update):**

- Add exercises 2.5.11, 2.5.106.
- Emphasize the main idea of subsection 6.4.3 to make it easier to pick out.
- Minor wording improvements throughout.
- Add a couple of sentences on how the arrows in vector fields are scaled in 3.1.4.
- Add a few explicit cross references instead of vague "in a previous example" and similar.
- Fix the errata.

**July 21st 2020 edition, Version 6.1 (edition 6, 1st update):**

Very minor update.

- Some minor improvements in language and style, and minor typo fixes.
- A few very minor clarifications.
- Mainly, fix the errata.

**November 7th 2019 edition, Version 6.0 (edition 6, 0th update):**

This is a major update. The main addition is a new appendix on linear algebra, so that a combined diffyqs/linear algebra course can be run with the book. There is also a new optional section on linear first order PDEs, and another on solving PDEs with Laplace transforms. A bunch of new exercises were added, some figures, and exposition was improved throughout.

Numbering is as
consistent as possible. **Exercises, chapters, sections were not renumbered** (as
always). Figures, numbered examples, some subsections, and equations might have changed numbers due to the edits
(see below).
The name of first and fifth chapter was updated (but not the number). So
the book is therefore a drop in replacement, and existing courses should not need to be updated
(unless giving page ranges, or referring to Figures or numbered examples).

Here is a complete list of changes:

- Add appendix A on linear algebra.
- Add section 1.9 on first order linear PDE and characteristics.
- Add section 6.5 on solving PDE with Laplace transform.
- Add appendix B with a more complete list of Laplace transforms.
- Renumberings:
- Figures in chapters 1, 3, 4, and 8
- Examples in 2.2, 3.1, 4.6, 7.1 (past the root test), 8.4
- Subsections in 2.2
- Equations in chapter 4

- Change name of first chapter to "First order equations" since we are adding a PDE section.
- Change name of fifth chapter to "More on eigenvalue problems" as really eigenvalue problems have already been started in chapter 4.
- Use newpx (Palatino) fonts in the PDF version. The line length has gotten slightly shorter to improve readability.
- The pages are on the other hand sligtly longer to save paper.
- Exercises with parts use the tasks package so they are not too cramped anymore.
- Make bottom margin smaller to save paper.
- All floating figures/tables now have a gray border to better distinguish them from surrounding text, and all insets are on the right.
- Make the box for boxed equations a bit rounder.
- Improve wording and make minor clarifications throughout.
- 0.2: Make the four fundamental equations a separate subsection, to emphasize and to make it possible to refer to it.
- 0.2: Add a very short example (0.2.2) on two constants and initial conditions.
- 0.3: Use a general form for the Newton's law of cooling, and name the old version exponential growth equation, which is more correct.
- 1.1: Add plot of \(\frac{1}{1-x}\) to show the singularity.
- 1.3: Add plot of an implicit solution.
- 1.3: Add figure with a graph for the coffee example.
- 1.6: A little bit more detail on drawing phase diagrams.
- 1.7: Compute the first two steps in the example with \(h=1\) explicitly.
- 2.1: Actually work out the reduction of order method and give an example.
- 2.1: Add another short example of linear independence of two functions.
- 2.2: Make the first couple a pages its own subsection for consistency.
- 2.2: Add a short footnote on the word "ansatz" and add it to index.
- 2.2: Make the \(y''+k^2 y= 0\) an explicit example, so that we can refer to it, and give the version of the solution with sinh and cosh.
- 2.2: Reword the paragraph on why doubled roots rarely happen, so that the reader doesn't think it can just be dismissed.
- 2.2: Add example of complex arithmetic.
- 2.2: In Exercise 2.2.3 label the parts abc... rather than just bullet points.
- 2.4: Improve the illustration of the pendulum including the forces.
- 2.6: Remove blurb about not memorizing the formula.
- 2.6: Add short note about what happens when the forcing frequency \(\omega\) goes to infinity.
- 3.1: Add linked tanks example.
- 3.1: Add example for changing a second order system to first order system.
- 3.1: Number all the examples.
- 3.1: Define more of the terminology for systems.
- 3.1: Split into subsections.
- 3.1: Add the Picard theorem.
- 3.2: Fix/standardize reduced row echelon form definition what we had allowed swapped rows.
- 3.2: Mention \(\det(A)\det(B) = \det(AB)\) since it makes sense when talking about invertible matrices, and is generally a useful thing anyway.
- 3.4: Rephrase Exercise 3.4.2 to be more specific.
- 3.4: For the complex eigenvalue/eigenvector equation mention that the bar of zero is still zero.
- 3.4: After the complex eigenvalue theorem, write down explicitly the general solution for a 2-by-2 system.
- 3.6: In Example 3.6.1, also solve for some initial conditions and add a figure of the result.
- 3.7: Add a quick example of a \(3 \times 3\) matrix with defective a eigenvalue.
- 3.7: Add some extra explanation of what to do with algebraic multiplicity 3 defective eigenvalues, and improve exposition on the higher multiplicity.
- 4.1: Expand slightly more on the linear algebra connection for eigenvalues/eigenvectors.
- 4.1: Add footnote with the definition of sinh, cosh, as some students might be coming to this sections not having seen these (or not having seen them recently), and may have seen the solution in terms of exponentials.
- 4.1: Add another picture of whirling string for a higher eigenvalue.
- 4.2: Add footnote with reference to the examples where \(x''+\lambda x=0\) is solved.
- 4.6: Add small example where no series is computed.
- 4.7: Add a little bit of intuition for the wave equation.
- 4.7: Update figure of the plucked string.
- 4.7: Add some discussion on what the solution says about the sound of a guitar. Also add some plots for fixed time of the shape of a plucked string.
- 4.8: make the verification computation easier by using the second form.
- 4.8: Add more comments on D'Alembert, why do corners persist, what about corners if we have to take second derivatives, and what D'Alembert says about influence of initial conditions.
- 4.8: Add hint to exercise 4.8.5.
- 4.10: A few clarifications.
- 5.1: Add explicit expansion of x to the example at the end.
- 5.2: Rename the section to "Higher order eigenvalue problems".
- 5.2: Add a footnote on the constant, and more detail to the derivation.
- 7.1: Add root test and an example for it.
- 8.1: Add a short paragraph on what happens near noncritical points to really justify why we are looking at the critical ones
- 8.4: Add another example and a figure for a limit cycle.
- 8.4: Add example for not simply connected domain with a closed trajectory
- Add exercises (other than in new secions/the appendix) 0.2.13, 0.2.14, 0.2.106, 0.3.103, 1.2.12, 1.2.106, 3.1.6, 3.1.7, 3.1.106, 3.5.5, 3.8.12, 3.8.105, 4.3.11, 4.3.106, 4.6.11, 4.6.105, 4.7.8.

**March 4th 2019 edition, Version 5.5 (edition 5, 5th update):**

*Pagination and numbering has not changed at all. The following changes were made:*

**Exercise 1.8.102 part c) changed to fix erratum.**- Add short discussion of fixed/free ends to 5.2 to fix erratum.
- Mention IODE as it is now supported again.
- Fix the errata.

**October 11th 2018 edition, Version 5.4 (edition 5, 4th update):**

*Pagination and numbering has not changed at all. The following changes were made:*

- A couple of minor clarifications in section 3.1
- All links change to https:
- Fix the errata.

**February 22nd 2018 edition, Version 5.3 (edition 5, 3rd update):**

*Pagination and numbering was kept as constant as
possible. The following changes were made:*

- Reorder the Notes on the Notes section to be more logical.
- A new web version based on PreTeXt which is much easier to browse, although it is a bit more network-demanding perhaps. This has been online since early January.
- A few fixes in readability and style, and a few typo fixes.
- The sources have been reorganized, moved to github, and split into chapters, hopefully making modification easier.
- Fix the errata.

**November 1st 2017 edition, Version 5.2 (edition 5, 2nd update):**

*Pagination and numbering was kept as constant as
possible. The following changes were made:*

- Add a very short explicit example of turning an equation into a system in 3.1.
- In 3.3, explicitly say that the fundamental matrix is if you have \(n\) linearly independent solutions.
- In section 3.6, add a sentence about when force is given, then needing to divide by the mass matrix. Similarly in the example, do that explicitly in the equation (it comes out to be the same vector in this example).
- In section 3.7, make the computation for the \(2 \times 2\) system a little bit more streamlined and make it clearer as to what should be solved as \((A-\lambda I)^2 \vec{v}_2 = \vec{0}\) is always automatically satisfied if \((A-\lambda I) \vec{v}_2 = \vec{v}_1.\) So only mention solving \((A-\lambda I)^2 \vec{v}_2 = \vec{0}\) as a remark afterwards.
- In the whirling string example in 4.1 mention units explicitly, and also mention the example of increasing tension on a spinning jumprope.
- At the end of 4.2, mention the terms "fundamental frequency", "spectrum", and "overtones".
**Exercise 8.4.103 changed slightly**to fix an erratum.- A few fixes in readability and style, and a few typo 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.

**March 21st 2017 edition, Version 5.1 (edition 5, 1st update):**

*Pagination and numbering was kept as constant as
possible. The following changes were made:*

- In Example 2.4.1, I committed the cardinal sin of taking arctangent of 1 on a calculator, so now it says \(\pi/4.\) Thanks to Erik Boczko for pointing this out.
- Exercise 2.5.7 has a typo leading to a different problem than intended. I removed the "What is going on?" seemed to imply that solutions were different, and was confusing to students. Instead of changing the exercise though, I added problem 2.5.10, \(y''-y=e^x\) where you get different solutions using undetermined coefficients and variation of parameters. Thanks to Asif Shakeel for pointing this out.
- A couple of very minor style improvements, and grammar fixes.
**Exercise 5.1.102 changed slightly.**There was a typo where the \(-2x\) should be \(+2x.\) The solution in the back is for this correct version of the problem. So I changed the \(-2x\) to \(+2x.\) I added a footnote that for a challenge the student can try the "wrong" version.- 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 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.

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 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.

**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.

**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 tongue 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.

**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.

**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

**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.

**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.

**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.

**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.

**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 grammatical fixes.

**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)

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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).

**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.