* Add section 1.9 on first order linear PDE and characteristics * Add section 6.5 on solving PDE with Laplace transform * Add appendix A on linear algebra. * Add appendix B with a more complete list of Laplace transforms * 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 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 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: In the exercises instead of just a quick note, actually work out the reduction of order method and give an example. * 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 3x3 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 * 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 * 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.