
Basic Differential Equations

Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time.  Sophus Lie
 Differential Equations Math 240 Notes Section 003 Nakia Rimmer University of Pennsylvania Fall 2007 (PG)These are supplemental notes for the second semester of undergraduate course in differential equations on systems of ODEs and an introduction to PDEs, primarily focusing on the linear algebra and vector calculus needed to support such a course as well as containing solutions to homework exercises. They aren’t as detailed and informative as Rimmer’s lecture notes on calculus, but they do have some useful stuff.
Ordinary Differential Equations W. E. Schiesser Lehigh University (PG) This is a oddball, purely applied undergraduate course on ordinary differential equations that combines basic solution methods with sophisticated computer programming methods to demonstrate how differential equations are studied and solved in real world situations. Extensive programming psuedocode is interspersed with detailed calculations using mostly nonlinear methods such as the Euler and RungeKutta methods, which usually aren’t studied at this level. Basic concepts such as order, boundary conditions and initial conditions and solution spaces are covered, but there’s virtually not theorems or proofs. None. Very strangemost applied notes at least state major theorems. Not here. Physics and engineering majors may find them useful, but for mathematics students, these notes will be helpful as supplementary reading only.
 A Short Differential Equations Course Kenneth Kuttler BYU January 11, 2009 (PG13) This really should be called “A Short Honors Course In Differential Equations” since it it’s part of Kuttler’s honors calculus course at Bringham Young that tries to combine theory and applications in equal measure, something I’m all for. As a result, it’s considerably more rigorous then the average beginning differential equations course. Unfortunately, Kuttler wants the course to be self contained, so nearly onethird of the notes are dedicated to a detailed review of the theory of calculus needed to support the course. I have mixed feelings about this. On the one hand, it does make the notes self contained for the reader and that makes them more accessible to someone just coming in who doesn’t necessarily have the background. It also contains some material that’s absolutely essential for differential equations but usually isn’t covered in the standard theoretical calculus courses, like the matrix exponential. But given how students taking such a course or using these notes would probably have most of this background anyway, it seems like a gargantuan waste of time and space to include this material. “These notes are for an honors course in DEs and require advanced calculus and linear algebra as prerequisites, lecture notes for which can be found at my website.” Done. Over. That major complaint aside, the remaining 64 pages actually on differential equations are an excellent presentation at this level. It covers all the basics: first and second order ordinary differential equations, initial conditions, characteristic equations, the Wronskian, systems of linear equations, power series expansions, singularities, etc. Chapter 4 also gives a brief account of the theory of ordinary differential equations beginning with the Picard iteration theorem, which is completely appropriate in a course at this level. Unfortunately, the notes have one major flaw that keeps them from being rated one of my best sources here: no exercises. None. Nada. It’s really sad, toothe quality of these notes is otherwise truly first ratebut I can’t in all good conscience recommend a source without exercises for students to chew on. Kuttler says it’s the draft of a book that’s not done yet. Ok, fine, but I don’t see how useful putting the draft up sans exercises is for any of his students as a stand alone source.
 Differential Equations I Math B44 Lecture notes by Peter Selick University of Toronto Fall 2009 (PG)This is a terrific and very complete set of lecture notes for a standard undergraduate differential equations course at the University of Toronto. It’s very similar to Kutz’s notes below in both content and level and Selick’s notes are also intended to supplement Boyce and Diprima. The main difference is that Selick’s notes are much more detailed with many more examples completely solved. They’re also beautifully written with many pictures and insightful comments. Both this and Kutz’s notes are must have supplements for any student taking undergraduate differential equations from this textthey’ll be immensely helpful for beginners in mastering the practical aspects of this allimportant subject. They’ll also be very useful for more advanced students who need a solid review of the basics
 before moving on to more advanced courses on differential equations.
 Differential Equations J.Nathan Kutz University of Texas (PG) An excellent undergraduate course on differential equations requiring no more then pencil pushing “practical” calculus to understand, much like it’s inspiration, the classic Elementary Differential Equations by Boyce and DiPrimawhich most of us first cut our teeth on for a first course in differential equations in my generation. The content of the lectures follows the book fairly closely: Linear and nonlinear equations. initial conditions, order, Taylor series. Laplace transforms, chaos and nonlinear solution methods, all with many examples from the physical sciences and engineering.The main addition to these notes over the texthook are linear algebra methods, such as eigenvalues and eigenfunctions, which are so critical to understanding not only basic differential equations, but the importance of the role of linear algebra in analysis in general. A very good source for students taking their first differential equations course.
Differential Equations Emre Sermutlu Camkaya University 2011 (PG)This is a standard undergraduate introduction to differential equations for engineering students that’s low on theory and high on pencil pushing calculations and examples. Not that that’s bad, mind youstudents have to be able to master these techniques before a theoretical presentation can fully be taken advantage of by them. Also, these lectures do have a few things that make them different from the run of the mill sort of notesthere’s an introduction to partial differential equations, Fourier series and many examples and exercises. But it’s not going to be useful to anyone who wants to use them for more mathematically inclined courses. Coupled with a more theoretical presentation lacking examples and exercises, such as Kuttler above, a pretty solid first course in differential equations for advanced students can result. But otherwise, these notes are basically good for pencil pushing practice and very little else.  Differential Equations Robert E. Terrell Cornell University (PG)This is a set of notes Terrell wrote to supplement his applied mathematics courses for engineers and as such, we shouldn’t expect too much in the way of mathematical rigor in them. (In fact, Terrell only states theorems without proof. How’s THAT for nonrigorous, engineering geared notes? ) What he DOES do easily is make the concepts live and speak to us through the motivating prose and the many concrete examples and graphics. Again, not really for mathematics students and it’s really for more practical people, but coupled with a more precise treatment, it can be very helpful.
 Differential Equations James S. Cook Liberty University Department of Mathematics Spring 2013 (PG13) This is a very nice, well written set of lecture notes for a first course in differential equations that balances theory and applications well. This course it pitched with minimal prerequisites, just pencil pushing calculus and a good course in linear algebra. There is an enormous number of completely solved problems, well written proofs of major general theorems appropriate in level for this course and many pictures of vector fields and level curves for the solution sets of the given equations. Cook explains things very well and provides a lot of motivation in addition to careful proofs when it’s appropriate. There are no exercises in the notes themselves, but Cook has written an extensive set of problem sets to accompany them, they can be found at the course homepage here. The exercises are well constructed and well written, but there really aren’t enough of them. The diligent student is going to need to supplement them with more problems, but that won’t be hard to do at this level as there are a ton of cheap sources of exercises for beginning courses. A very good asset indeed for a first course and another good job by Cook.
 Differential Equations Mark Pedigo University of Washington St.Louis 20122013 Fall Semester (PG) This is a standardand I do mean standard –undergraduate course in differential equations at the University of Washington for students with a background in calculus only with some brief supplementary notes by Pedigo. They’re well written, readable and do have some original stuff you don’t usually find in a course at this level, like the Bernoulli equations. But otherwise they’re the usual run of the mill undergraduate DE course and nothing special. You’re better off using Finan’s or Selick’s notes as your main textbook.
 A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University March 12, 2012(PG)
 A First Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University(PG)
 A Second Course in Elementary Ordinary Differential Equations Marcel B. Finan Arkansas Tech University (PG) This online textbook was written and published in 2 halves, corresponding to both semesters of the yearlong undergraduate course in differential equations at Arkansas Tech. The polished nature of the bookdetailed organization and modularization, extensive exercise sets and of course, it’s size and scope clearly make it a book as opposed to merely a set of lecture notes. Hopefully Finan will allow it to remain online for free access for the foreseeable future because it’s the kind of book you wish you’d had as a student first taking the subjectverbose and clear, concrete and mostly practical yet doesn’t avoid mathematical rigor and proof at the level appropriate for the intended audience. That audience is clearly beginners i.e. students who haven’t yet been exposed to serious mathematical analysis and have a very good working knowledge of “practical” calculus of both one and several variables. In the second course, a great deal of computational linear algebra is used to develop aspects of systems of linear differential equations, but since all matrices have real valued entries, all matrix algebra is developed as needed and no abstract vector space theory is used. This makes the course an excellent precursor to linear algebra courses as everything used here can be used as examples in such a followup course. I have mixed feelings about this last part, I think linear algebra should at least be taught concurrently with a first course in differential equations. But I suppose it’s a small price to pay for such a wonderful course book and you can always supplement it with more rigorous treatments like Kuttler. Think Boyce and Diprima, only free and with better graphs and pictures and more sophisticated material, like the matrix exponential. A beautiful, comprehensive book I fully intend to use for my students when I teach baby differential equations for the first time and one I intend to use over and over for my tutoring.
 Differential Equations M.T. Nair and YVSS Raju INDIAN INSTITUTE OF TECHNOLOGY MADRASMA 2002 (PG) These are the notes and exercises for a complete undergraduate course on both ordinary and partial differential equations at IIT written by the three authors. It’s a bit more advanced then U.S, coursesit only gives a brief review at the beginning of first order ordinary differential equations, assuming they are known to the students, presumably from their calculus courses. So the focus of the first part of the notes is on second and higher order homogeneous and nonhomogeneous linear differential equations. It also includes a fairly detailed introduction to partial differential equations. including discussions of first and second order linear PDEs and classification of the 3 major classes of PDEs:elliptic, parabolic and hyperbolic and some basic solutiom methods. The notes are very clear, but rather concise. They are sparse with examples but provide many proofs and insightsthe few examples they do have are discussed in exhaustive detail. A good choice if you’ve got a strong background in calculus and linear algebra and you prefer more of an austere presentation then either Finan or Selick’s notes give. You can also use them as a supplement to those courses to provide more rigor they they have.
Techniques in Ordinary Differential Equations Kevin Mirus Madison College, Fall 2010 (PG) This is yet another differential equations course for undergraduates based on Boyce and Diprima requiring very little background and emphasizing problem solving over theory. These notes are exactly what the title says they aresome supplementary notes focusing on the techniques of solution learned in the course with some added practice. I found them a bit sloppy, but containing some good graphs, calculations and “bullet points” on the key ideas studied. Frankly, I think The Schaums’ Outline on Differential Equations is much better as a study aid, but since they’re free, why not? You may find them useful come test and quiz time.
Differential Equations  Mathematics  MIT OpenCourseWare (PG) Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.  Differential Equations and Computer Methods Queen’s University Mathematics and Engineering and Mathematics and Statistics (PG13) These are the supplementary notes for the standard undergraduate course in differential equations at Queen’s University of London. quite a bit more advanced and much more tersenearly all of the proofs of the major theorems are left as exercises and there are virtually no pictures.There’s also material on stability theorems, nonlinear equations, the Laplace transform and numerical methods of solution. I think the goal for these notes was for the professor to fill in the details in class. Also, they contain nothing on PDE’s and require linear algebra as a prerequisite. These notes clearly are too terse to be used for self study by any but the very strongest, gifted studentseven good students I think would struggle with them since they’re so terse. They could certainly be used for review by students who have mastered a stand ard course covering this material or by instructors who are prepared to fill in all the gaps. As I’ve said, though they’re rather dry.
 Differential Equations Jeffrey R. Chasnov The Hong Kong University of Science and Technology (PG13)Very similar in content to Nair and Raju’s notes above but much more detailed with many examples, computations and derivations, as well as proofs of most theorems. Well written with many full phase space diagrams, which are so important for fully understanding the behavior of solution spaces. There are also many good exercises and applications including some not commonly tackled, such as damped resonance and a lengthy and well written chapter on bifurcation theory. There is also a series of video lectures to accompany the lecture notes, with the link in the notes. More advanced then most courses at this level. An excellent source for serious students of differential equations, well worth checking out.
 A Proposed Text for Differential Equations Kenneth Howell University of Alabama in Huntsville (PG)This is an online textbook on differential equations that’s still a work in progress, again at the basic postcalculus course level. On the surface, it looks like a boilerplate set of notes that runs the usual cast of characters for such a course: first order equations, seperability, integrating factors, solution space vector fields, second order equations with standard solution methods like reduction of order and variation of parameters, yadda, yadda yadda. And with so many similar sources available online, the selfstudy scholar might be sorely tempted to skip on it. And in doing so, he or she would be making a MAJOR mistake. This, to me.with the possible exception of Finan, is the single best free online source for learning differential equations from scratch. No lie. Yes, Howell does cover all the standard topics and he does it without serious rigor, as most texts at this level do. But what sets his text apart is the enormous conceptual depth and originality of examples with which he discusses each topic. One example is chapter 8, where he gives the best discussion of integral curve sketching I’ve ever seen in terms of “slope fields” and the procedure which constructs them. Without stating it in those terms, he constructs the tangent vector field to the level surfaces of the solution spaces and the procedure he gives is quite clever and requires only basic calculus. He gives a very large number of examples and their graphs amd uses these examples to give an elementary discussion of solution stability.Such a discussion would be enormously helpful for motivating stability theory in a more advanced, subsequent course. Another example is Howell’s creative spin on the standard elementary modeling with differential equations in chapter 10: he considers a rabbit farm as his central example and how to model the population growth of the rabbits on the farm over 5 years. This can be done a variety of ways, most famously using Fibbonaci number sequences, but it can certainly be done using first order exponential growth, which is what Howell does. Another example is how he has an entire chapter on the convolution function method in Laplace transform problems, again a topic usually reserved for more advanced courses. This is an outstanding book on differential equationsdownload the preliminary version before the major publishing houses see this draft and a bidding war ensues that end up costing you 200 dollars to use it. UPDATE: Howell apparently has published a preliminary 4 chapter hardcover version of the book for further classroom testing at UAH through HaydenMcNeil Publishing ,while continuing to revise and polish the book, currently in the 20142015 edition. I think this means the book is nearing "official" publication. The new version appears to have the same general layout, but more details. If anything, the book has improved and I even more strongly encourage teachers and students of differential equations to look at it.
 Ordinary Differential Equations and Control Semester Lecture Notes on ODEs and Laplace Transform (Parts I and II) J D Evans and E.P Ryan University of Bath October 2012 (PG13)This is set of notes in 2 parts for a more sophisticated course on differential equations, presuming some advanced calculus and linear algebra. The notes and their associated problem sets cover standard topics in ordinary differential equations and dynamical systems, including linear autonomous equations, first order systems, heterogeneity and homegenity, the Laplace transform and the convolution integral and basic control and stability theory. Terse but readable with many good problems. For strong math students only.
 Differential Equations and Linear Algebra Lecture Notes Simon J.A. Malham HeriotWatt
 University (PG13) Excellent set of lecture notes at the upper level undergraduate level combining basic differential equations with a first course on abstract linear algebra. All the usual topics you’d expect in such a course are herelinear second order ODEs, oscillator theory and Hooke's Law, general ODEs and their classification,homogeneous linear ODEs ,the principle of superposition , nonhomogeneous linear ODEs, linear operators, the method of undetermined coefficients , initial and boundary value problems, the Laplace transform and much more. Standard introductory topics on abstract linear algebra are covered towards the end, including GuassJordan elimination, eigenvectors and eigenvalues. However, these notes are clearly pitched at a level where a bit more mathematical sophistication is needed for then the usual pencil pushing calculus course in the US. But that’s generally true of lecture notes from courses in Europe and the U.K in particular. Well written, lots of examples and graphs and a good selection of topics for a course geared at serious mathematics majors. Overall, a very good choice for mathematically strong students.
 Differential Equations and Linear Algebra Lau chi Hin Chinese University of Hong Kong (PG13)It’s all the rage now in academia to combine first undergraduate courses in differential equations and linear algebra into a single year long course. This makes sense in a pedagogical waylinear differential equations and their solution spaces form excellent and important examples of vector spaces of linear transformations and their structures. This is an excellent set of lecture notes for such a course,covering in a unified manner the standard theory of differential equations and a concise, yet rigorous presentation of the elements of linear algebra Linear first and second order ODEs, oscillator theory and Hooke's Law, general ODEs and their classification,homogeneous linear ODEs ,the principle of superposition , nonhomogeneous linear ODEs, the method of undetermined coefficients , initial and boundary value problems,. linear systems and matrices, row and reduced row echelon forms, elementary row operations, inverse and identity matrices, determinants, vector spaces, linear independence and bases, dimension, row and column spaces, rank and nullity, eigenvalues and eigenspaces, linear transformations and their relationship to matrices, orthogonal maps and spaces. The treatment of linear algebra is surprisingly complete, including a proof of the CayleyHamilton theorem. Rather unsurprisingly, it emphasizes diagonalization and spectral theory, which is so critical in the theory of linear DEs.Quite well written with lots of examples while keeping the preparation level limited to a year course in nonrigorous calculus. An very good source for both kinds of courses, particularly for mathematics majors due to the careful rigor of the notes. Highly recommended.
 Linear Algebra and Differential Equations Matthew D. Johnston Van Vleck University of WisconsinMadison Spring 2013 Course Materials (PG) Another course combining standard first courses in differential equations and linear algebra. This one is less theoretical then Kapitula and focuses much more on computational and solution methods relying on the technology of linear mappings. This is reflected in their very different structurethe first 60 or so pages is a computational course in ordinary differential equations covering the standard topics (order, first order equation solutions, the method of undetermined coefficients, variation of parameters, etc) before moving on to an applied linear algebra course emphasizing those aspects needed in the solution of differential equations, such as matrix operations and spectral theory. There are very few proofs, but many examples and step by step computations that will be quite helpful for the beginner. I don’t think one should use it as a sole textbook, but it will prove helpful as a supplement. Recommended.
 Linear Algebra and Differential Equations Worksheets University of California at Berkeley 2012 edition: (PG13) Exactly what the title says it isa very extensive collection of exercises intended to supplement the undergraduate course in differential equations and linear algebra at the University of California at Berkeley. The purpose of these exercises is to provide intuitive and computational supplement problems for the students in this very dense and usually quite theoretical course, which is usually taught out abstract textbooks such as Axer and Hirsh/Smale/ Devaney. The result is an outstanding and diverse collection of problems that runs the gamut from computational plug and chug problems to rigorous proofs to constructing sophisticated examples in both linear algebra and physical problems in differential equations. The scope and diversity of the problems can be inferred from the table of contents: Introduction to Linear Systems . Matrices and Gaussian Elimination The Algebra of Matrices......Inverses and Elementary Matrices..Transposes and Symmetry... Vectors..\General Vector Spaces Subspaces,Span,and Nullspaces.... Linear Independence.. Basis and Dimension... Fundamental Subspaces and Rank... Error Correcting Codes...Linear Transformations. Inner Products and Least Squares Orthonormal Bases. Determinants Eigenvalues and Eigenvectors. Diagonalization Symmetric Matrices Wronskian and Linear Independence ..Higher Order Linear ODEs .Homogeneous Linear ODEs Systems First Order Linear Equations. Systems of First Order Equations–Continued. Oscillations of Shock Absorbers Introduction to Partial Differential Equations Partial Differential Equations and Fourier Series.Applications of Partial Differential Equations A wonderful collection that any student or teacher of either subject should definitely add to his or her toolbox. Very highly recommended.
 Differential Equations Arthur Mattuck Notes and Exercises (PG) This section includes PDFs with supplementary notes and exercises files. Some nice, brief notes and exercises that are bit more mathematically serious then this level of course normally allows, including complex numbers, linear operators and an introduction to stability conditions.Far briefer and less comprehensive then Haynes Miller’s related notes for the same course, but a good supplement from a master teacher of analysis that will help beef up any “applied” intro to differential equations with some real mathematics suitable for a course at this level. Contents Definite integral solutions G Graphical and numerical methods C Complex numbers IR Inputresponse models O Linear differential operators S Stability I Impulse response and convolution LT Laplace transform LS Linear systems of ODE's  LS1 of 6 , LS2 of 6 LS3 of 6 LS4 of 6 LS5 of 6 LS6 of 6 GS Graphing systems LC Limit cycles Exercises 1 Firstorder ODE's Higherorder ODE's 3 Laplace transform 4 Linear systems 5 Graphing systems 6 Power series Fourier series Solutions to exercises
 Differential Equations K. Harris University of Michigan Fall 2008 : (PG)This is set of Powerpoint turned PDF lectures to supplement a standard undergraduate differential equations course based on Edwards and Penney. This course has an applied focus and the supplements share the same flavor, they focus on solution methods and concrete calculations of the various forms of ordinary differential equations. So if you’re looking for a hard presentation of the theory of differential equations, look elsewhere.But in the case of differential equations, a mastery of the calculational methods is just as important if not more sothen understanding the underlying theory. These notes give many good and important examples and graphs of solution spaces as well as discussions of stability and bifurcation problems. I wouldn’t use them as a sole source, but as a source of examples to balance a theoretical presentation like Kuttler, it’ll work very well. It’ll also be handy come test time for review and additional drill.
 Differential Equations spring 2013 lecture page Nick Kovasar University of Utah (PG)This is the lecture page of Math 22501 at the University of Utah, for the spring term 2013. These notes are very similar to the Harris supplementary notes above in content, but there’s almost no discussion and they consist almost entirely of solved problems and graphs. Not that that’s bad, mind you, examples are always a good thing to have a stock of. But there’s even less meat here then in Harris’ notes, they can’t be used as anything but a supplement to a good set of notes that lacks examples, like Mulham’s notes or Coddington’s textbook. For supplemental use only.
 Differential Equations II Richard Bass University of Connecuit Fall 2012 (PG)These are course notes for a second semester of a standard differential equations course.They cover second order linear ordinary differential equations, power series methods of analyais, the one dimensional oscillator, an introduction to partial differential equations via the wave equationwell, you get the idea, the usual cast of characters. They’re written with Bass’ usual clarity, but nothing original or inventive hereand the notes come with no exercises, which really limits their usefulness. Ok for use in a course of this level, but really nothing special.
 Differential Equations Amol Sasane Department of Mathematics, London School of Economics (PG13) This very strong and comprehensive set of notes is designed for the standard differential equations course at the reknowned London School of Economics, which is also well known for its excellent mathematics program. Although the specific prerequisites for the course appear to be no more then calculus and a good course in linear algebra, it’s also clear the level of rigor and care taken in the notes indicates that students with no exposure to careful proofs are going to struggle with the level of sophistication. In the U.S.. the notes would probably be more suitable for students with either a strong honors calculus or an advanced calculus/elementary real analysis course under their belts. There are many explicit e xamples and calculations and in particular, there’s a great emphasis on the theory and construction of phase diagrams one usually doesn’t see until more advanced courses.Singularities, diagonalization analysis and stability aspects such as the Lypanov Theorum are discussed in great detail and at an elementary level. There are also lots of nice exercises, none easy but none too hard, either. I wish I’d had these notes when taking advanced ordinary differential equations An excellent resource, one of the best I’ve seen and a great choice for learning intermediate level differential equations or reviewing the material before graduate school .
 Elementary Differential Equations Ovidiu Costin Ohio State University Math 415 (PG)A set of notes foryou guessed it!the postcalculus level routine elementary differential equations course. Actually,they’re not a full course set of notesthey cover just the materials of the second half of the course, after covering the first 3 chapters of Boyce and Diprima (up to and including separable differential equations). The notes cover, in order, nonhomogeneous equations: method of variation of parameters,vibrations with and without damping and forcing, power series and examples of series solutions near regular points, twopoint boundary value problems, Fourier series and convergence theorems, Fourier series for even and odd functions and it ends with full discussions of the heat equation with the classification of boundary conditions ( both zero and nonzero) as well as the wave equation, D’Alembert’s solution and Laplace’s equation. The Powerpoint presentation is quite effective for a class like this because it allows the calculations and drawings of each exampleand there are manyto unfold step by step as the student turns the “pages” of the PDF format. Costin is careful, but very concrete. A student will get a lot out of these very readable and detailed notes as a supplement to the standard coursewithin the limitations of what they’re intended to achieve.
 Introductory Notes in Ordinary Differential Equations for Physical Sciences and Engineering Marcel B. Finan Arkansas Tech University (PG)This is another of Finan’s many wonderful online textsthis one is aimed at engineers and other applied mathematics students. Unlike his much longer 2 semester differential equations course for mathematics undergraduates, these notes are designed for engineering students and focus entirely on applications. Well, not entirely Finan states the critical theorems and gives motivations, but not proofs. Basically, this book is the union of the 2 semester courses above with all the rigorous proofs deleted. It preserves all the many positive qualities of that more comprehensive work while changing the focus entirely to applications and intuitive arguments. As I’ve said elsewhere, I don’t approve of having distinct courses for pure mathematics students and applied mathematicswhich I ultimately consider all the physical sciences to bestudents. Depriving applied students of training in abstract thinking I think really hinders their problem solving ability when it comes to previously unknown or original problems. Mathematics really teaches you how to create a solution where none existed beforeand it does that by training you to analyze a situation completely from scratch and constructing your own definitionsand making deductions from those definitions. Sadly, I’m the minority in this opinionwhich is why Finan feels the need to write 2 books instead of just one for everyone involved. Still, it’s nicely written and filled with insights. So if you’re a engineering of physical sciences student who can’t stand logical proofs, give this a try.
Differential Equations for Engineers by Jierí Lebl University of Wisconsin UrbanaChamplaign 2012 (PG)These notes in particular have become very popular and it’s easy to see why. Like most mass market books on differential equations geared towards applied students, the rigor to application/computation/example ratio is within an epsilon of 0 ( i.e. very small). There are a ton of detailed examples from physics and engineering as well as many diagramsagain par for the course in books for this intended audience. What’s different here is that Liebl isn’t afraid to mix simple “calculus level” proofs in with the motivation and computations, as long as he keeps it as simple as possible. For example, he proves that any solution of the form e^{nx } where n is an integer is linearly independent without Wronskian computations. Because of this “baby proof” approach, the book is extremely versatile and can be used for all kinds of undergraduate courses in differential equations. It can easily be used as a text for a first course in differential equations for mathematics students with minimal backgrounds in high school or college freshman. Courses for stronger students can also use it as a supplement to theoretical presentations, to provide applications and motivation. The writing is lively, literate and very informative. In short, this is a fine book that will prove useful as an introduction to differential equations for all kinds of students. Highly recommended.  Differential Equations by Erin Pearce Cornell University (PG) Yet another set of lecture notes for the standard postcalculus differential equations course. The thing about these particular notes that impresses me is the excellent organization of the content. Most lecture notes have at least some structural issues, which is one of the things that distinguishes them from actual textbooks, which can’t be published usually without getting most of these deficiencies worked out. Pearce’s are an exceptionthey’re wonderfully structured with theorems interwoven deftly with exceptionally detailed examples, direction vector field and solution curve graphs. He also includes many applications that are important but usually omitted from a first course, such as Newton’s law of cooling, resonance and centripetal force. They’re brisker in pace and not quite as comprehensive as Finan or Howell, but better organized and just as accessible and useful. An excellent source for this course.
 Differential Equations Problems, Answers, Handouts Joel Feldman University of Britiish Columbia (PG)Yet another set of supplementary notes for the plain vanilla undergraduate postcalculus differential equations course. Their purpose appears be to provide some additional material and exercises not in the textbook, such as linear regression and a list commonly used Fourier expansions for calculations. Feldman is a good teacher and provides some helpful, if not musthave material for the course. Worth checking out for what it is, but again, don’t expect magic.
 DIFFERENTIAL EQUATIONS Garrett Etgen University of Oaklahoma Spring 2013 (PG) Ok, now we’re talking about something substantial. As I said, I love when a mathematician takes the time and care to write up their own course materials entirely and posts them onlineit shows a real concern and love for teaching. Etgen’s notes are a perfect example of why this is a great practicehis notes assume a good grounding in linear algebra in his students, which allows him to pitch the course at a somewhat higher level then the usual boilerplate postcalculus DE course, but keep the rigor level beneath that required for an intermediate level ordinary differential equations course. It’s very difficult to find a standard text at this level and even if you did, it almost certainly wouldn’t have close to the content your personal vision of the course contains. He covers all the standard topics in such a course with lots of terrific examples, detailed calculations using elementary calculus,matrix manipulations and graphsyou can cut and paste most of the topics list from just about any other book at this level. His logic is just a little strange here. Basically his approach seems to be anything that needs more then pencil pushing calculus and/or linear algebra to explain or prove, omit from the course.This leads to some annoying choices on the author’s partfor example, he refuses to even state the existence and uniqueness theorem for first order differential equations, but happily does it for second order equations. He doesn’t prove it in either case, but we wouldn’t expect him to in a course at this level. But he could state these theorems and provide some motivation why they’re true. I understand the reasoning, but he could state some of the more important results without proof and give references. This helps picque the serious students’ interest for future courses. Still,this is just a minor quibbleotherwise, this is one of the best sets of notes
 currently available and would serve very well indeed as the basis for a beginner’s course. Indeed, coupled with Please’s notes whose link immediately follows, they could serve as a basis for a year long honors course on differential equations which mixes theory and applications in equal measure.
 Differential Equations Colin Please, based on notes by Paul Tod MT 2012  Mathematical Institute  University of Oxford (PG13) These are the 2012 lecture notes and exercises for the first course in differential equations at Oxford University. As such, it requires a full command of both advanced calculus and linear algebra to be fully understood and so will serve American students as an honors course only. That being said, it’s another terrific online source from one of world’s great universitiesa beautiful course for strong students in the Oxdord tradition: Concise but complete, rigorous but clear, modern yet inititive with many diagrams and graphs. While the emphasis here, as expected, is on mathematical rigor and there aren’t as many examples as in the previous sources, it’s one of the best written and tightly structured sets of lectures currently available. Coupled with a more applied treatment such as Finan or Etgen, Please’s notes will provide the basis for as comprehensive a course as one could possibly design for both students and selfstudy. A must have for those serious about learning differential equations right the first time
 Differential Equations Mark Pedigo Washington University of Saint Louis (PG)A fairly standard set of lecture notes for a fairly standard postcalculus differential equations course. They’re nicely written, but there’s nothing here you won’t find explained just as well or better in the previously listed sources in this bibliography. There IS something worth checking out at this site, though Pedigo has some very nice practice quizzes and tests free for download. The problems are well thought out and insightfulstudents definitely should use them in their reviewing and studying. But other then these problem supplements, nothing special here.
 Differential Equations Shahriar Afkhami Virginia Tech University (PG) Yet another applied first course in differential equations. In quality, these are a notch above the ubiquitous note sets online for courses of this kind, although not quite as good as the superior candidates like Finan, Selick, Jiebl or Howell. These notes are most similar in spirit to Etgen’s notesalthough not quite as substantial or readable in my opinion. They emphasize detailed computations and examples, as well as the use of linear algebra in solution methods. They also contain many examples with good graphs illustrating virtually all of them. The main original contribution these notes make is to use MATHLAB to construct direction fields and numerical solutions. But this really doesn’t make a major differencecomputer algebra systems are now standard procedure in differential equations course prep. Still, it’s a quality presentation amd the examples are more detailed then usual. You could do a lot worse if you’re struggling in your differential equations course then to download and study Afkhami’s notes.
 Differential Equations Mathematics African Virtual University (PG) This a “book” for an online course in differential equations in Africa. It’s an unusual and useful addition to the canon of notes It’s fairly standard in it’s choice of topics for such a coursealthough it’s more comprehensive then most and gives a fairly throrough presentation of the basics of both ordinary and partial differential equations.It focuses largely on examples and methods of solutiongiving a ton of very detailed examples and computations, as well as many exercises for the student to mull over. However, it doesn’t shy completely away from proofs like most books at this level doall the main results that can be proven by just calculus and some basic matrix algebra are given full proofs. It’s most similar to Pearce’s notes, but more basic and not as well organized or detailed. Think The Schaum’s Outline with references. Still, a handy source, although not as good as the more comprehensive sources like Leibl or Finan.
 Differential Equations J.E.Lindey MTH4102 Queen Mary University of London (PG)A fairly standard,applied plug and chug introductory course in differential equations from across the pond. This must be the course for the applied and engineering majorsat UK universities, such courses for mathematics majors, as we've seen, tend to be far more theoretical and rigorous.The prerequisites are the basics of matrix algebra or an applied course in linear algebra and nonrigorous calculus. Contents: difference quotient, derivative and linear approximation, geometric meaning of derivative, antiderivative, verification of solutions by substitution, solution by integration, geometric meaning of a first order differential equation, initial value problem, solution by geometric method, direction field, separation of variables for first order differential equations, implicitly defined solutions,first order linear differential equation, homogeneous and inhomogeneous equations, variation of parameters.Differential forms, integral curves, exact differential equations, integrating factors, homogeneous differential equations.linear second order differential equations with constant coefficients, homogeneous equations, superposition, characteristic equations, real roots, complex roots, degenerated roots,inhomogeneous equations with constant coefficients, method of undetermined coefficients, variations of constants formula, forced oscillations and visualisation,matrices, eigenvalues and eigenvectors (2dimensional),linear systems in two dimensions, equivalence to second order equation, solutions by linear algebra . phase space in two dimensions, solutions of linear two dimensional systems, stable/unstable nodes/foci, planar phase space portraits, classification of equilibria. autonomous equations and stability, linearisation of nonlinear systems, characterisation of equilibrium points. Some nicely written handwritten notes and diagrams and lots of good computational problemsbut for students looking for something rigorous or inventive, look elsewhere. Good for computational practice and review, but not much else.
 Differential Equations Math Haynes Miller MIT Spring 2010 (PG) This is an extensive lecture notes written to supplement the intensive elementary differential equations course at MIT. It is the second set of lectures for this course, the first set, written by Arthur Mattuck, we reviewed here. The primary audience for this course are applied mathematics majors and engineering majors and therefore the emphasis is on practical methods of solution. Since Miller and Mattuck are both exceptional mathematicians and moreover, Mattuck is a legendary teacher of mathematics, one would expect more mathematical substance than that description would imply. And you'd be right.While neither note set is exactly what one would call rigorousit really isn't supposed to be in a course of this nature nothing is pulled out either author's anus without careful motivation or proof, either. All the standard techniques of solving first and higher order ordinary differential equations are given, along with many practical applications to geometry, physics and other disciplines. Miller in particular likes to use the accruing interest in a bank account or credit card as a motivating model in many of his discussions. There are also many nice diagrams and exercises to supplement the textbook. It should be noted niether of these notes are really designed to function independently as a textbook, but to supplement a standard text.In that capacity, they're quite readable,insightful and will make nice bathroom reading. But as I said, they can't really function as a textbook. Fortunately, there are a few excellent cheap Dover paperback texts or online textbooks that are available free that would go very nicely with these notes.Recommended as a supplement. Contents (Miller) Preface Notation and language Modeling by first order linear ODEs Solutions of first order linear ODEs Sinusoidal solutions The algebra of complex numbers The complex exponential Beats RLC circuits Normalization of solutions Operators and the exponential response formula Undetermined coefficients Resonance and the exponential shift law Natural frequency and damping ratio Frequency response The Wronskian More on Fourier series Impulses and generalized functions Impulse and step responses Convolution Laplace transform technique: coverup The Laplace transform and generalized functions The pole diagram and the Laplace transform Amplitude response and the pole diagram The Laplace transform and more general systems First order systems and second order equations Phase portraits in two dimensions
 Linear Algebra and Differential Equations Harvard University Fall 2004 Oliver Knill (PG) (Another one of Knill's extremely terse, almost bullet point presentation notes of the elements of both linear algebra and differential equations. As terse as they are, they aren't as dry as some of his othersin fact, some of the discussion is rather amusing in a nerdy way.For example, check out his "transformation" of Arnold Schwarzzeneger in the section on linear transformations. There are also surprisingly many applications to the physical and social sciences. That all being said, they're very intensive,concise and cover a lot of material very quickly. I'm not sure the average student could use them for self study for a text. But for strong students who want to learn rapidly and desire a challenge, these notes and the accompanying exercises will do very nicely. Recommended for strong students. (Contents) 1 Introduction to linear systems 2 Linear transformations 3 Linear subspaces 4 Dimension 5 Orthogonality 6 Data
 fitting 7 Determinants 8 Diagonalization 9 Stability and symmetric matrices 10 Differential equations 11 Function spaces 12 Partial differential equations 13 Review and Vacation
 Differential Equations Jin Wang Old Dominion University Fall 2009 (PG)Very standard supplementary notes for an equally standard, plug and chug computational introduction to differential equations.Not bad,but again,nothing new to see here.Suggest you take a look and see if you like the presentation. Otherwise, move on. I wasn't impressed.
 Differential Equations I Birne Binegar Oaklahoma State University Summer 2014(PG) Differential Equations II Birne Binegar University of Oklohoma Fall 1998 (PG) Yet another very solid set of course notes and materials from Professor Binegar, these on undergraduate differential equations.Neither set of notes is mathematically rigorous in the sense of advanced calculus or abstract algebra, but they are both quite careful and do very little by "handwaving", The first is for an elementary course on the methods and theory of de's which he just completed giving over the summerit is essentially identical to a course he's been giving at OSU for many years and the lecture notes he first posted on the internet in 1999.They are,like all Binegar's lecture notes, very clear and concise and unafraid to give proofs to even "nonmathematics major" students. They cover all the major basic methods of solution for differential equations with examples and many insights. What's especially helpful about his notes is that Binegar often explains things that are usually not fully explained or understood by students in a basic calculus course For example, he takes the time that most calculus teachers don't to explain the difference between an indefinite and a definite integral on a closed and bounded interval. There aren't as many examples as I'd like, but it's important to keep in mind these are intended as supplementary notes and not an actual text. There are also applications,although not as many as I'd prefer, including initial value problems in projectile motion. There are many good exercises as well. The second course notes continue this style and approach for an advanced course with the prerequisite of a standard introduction to differential equations and a standard linear algebra course. The role of
 linear algebra in the theory of differential equations is of course critical and the point of these notes is to drive home to the mathematics student this importance,especially for systems of several variables such as partial differential equations. Stability of solutions, direction fields and eigenspace methods are emphasized, with many vector field computer generated graphs. Complex numbers are also utilized throughout. Matrix and linear mappings are used throughout. There are many more applications in the second coursewhich makes sense since in the real world, actual models have more then one variableincluding population dynamics and the general wave and heat equation solutions. There is an excellent introduction to Fourier series and their many applications to partial differential equationsincluding a nonrigorous but careful introduction to distributions through the Dirac delta function. Overall, this is another winner from Binegar. It will need to be supplemented by a textbook for more examples and exercises, but as I said earlier, there are many cheap ones available now. Together they'll serve for an excellent set of notes for a one year course on differential equations. Highly recommended. Contents ( DE I) Lecture 1: Introduction Lecture 2: Solutions and Classification Lecture 3: Graphical Methods Lecture 4: First Order ODEs and the Fundamental Theorem of Calculus Lecture 5: Separable Equations Lecture 6: First Order Linear ODEs Lecture 7: Constants of Integration and Initial Conditions Lecture 8: Exact Equations Lecture 9: Integrating Factors Lecture 10: Change of Variable Lecture 11: Second Order ODEs; General Theory Lecture 12: Reduction of Order Lecture 13: Second Order Linear Equations with Constant Coefficients Lecture 14: Euler Equations Lecture 15: Nonhomogeneous Second Order Linear Differential Equations Lecture 21: Higher Order Linear ODEs with Constant Coefficients Lecture 22: Sample Exam 2 Lecture 23: Solutions Via Power Series Lecture 24: Manipulating Power Series Lecture 25: Solving Differential Equations via Power Series Lecture 26: Differential Equations with Polynomial Coefficients Lecture 27: Singular Points and Convergence of Series Solutions Lecture 28: Series Solutions about Singular Points Lecture 29: The Laplace Transform Lecture 30: Laplace Transform and Initial Value Problems Lecture 31: Laplace Transforms and Piecewise Continuous Functions Lecture 32: Systems of First Order ODEs Lecture 32: Sample Final Exam (DE II) Lecture 1: Linear Algebra and Matrices Lecture 2: Homogeneous Linear Systems of ODEs Lecture 3: Nonhomogeneous Linear Systems of ODEs Lecture 4: Stability of Autonomous Systems Lecture 5: Interacting Species Lecture 6: Liapunov's Second Method Lecture 7: Fourier Series Lecture 8: The Heat Equation Lecture 9: The Wave Equation Lecture 10: Laplace's Equation Lecture 11: SturmLiouville Theory Lecture 12: SturmLiouville Theory and Nonhomogeneous Boundary Value Problems Lecture 13: SturmLiouville Theory and Special Functions Lecture 14: Green's Functions Lecture 15: The Delta Function Lecture 16: Numerical Methods Lecture 17: Numerical Differentiation Lecture 18: RungeKutta Methods Lecture 19: MultiStep Methods
 Differential Equations Nick Kovasar University of Utah Spring 2013 (PG)Yet another supplementary set of lecture notes for a standard introduction to differential equations. These are a bit more detailed then usual, although not as comprehensive as online texts on this material like Finan or Zwick. The required text for the course is the very popular Differential Equations and Linear Algebra by C.Henry Edwards and David E. Penney The notes are basically a condensed version of the text with some additions.They review the major definitions and applications in the textbook and give additional substantial exercises and examples, some utilizing MAPLE computations. The notes do add some additional concepts and applications that aren't in the textbook, such as Torricelli's law of fluid draning, Kirchoff's laws of current flow in electric circuits, population dynamics and many more. The notes are dry, but quite clear and brusque. Also, since linear algebra is not a required prerequisite, the needed matrix arithmetic and algebra are developed in the notes. There are also several very good sections on numerical methods of solutions and the use of MAPLE to solve them. There are also many good computer generated graphs. Overall,they do contain enough detail to be used by themselves as a course text, but they don't contain enough exercises by themselves. Still, coupled with a cheap textbook for a first differential equations courseand as I've said, a number of good choices for such a text exist nowthey will certainly make a solid,if not spectacular,course text for a first course in DEs. Recommended.
 Linear Algebra and Differential Equations Robert Winters Jeremy Marcq Harvard University 2014 (PG13) Here's yet another version of the combined linear algebra and differential equations course at Harvard. Winters' lectures are more substantial and detailed then the other versions of this course commented on here.He emphasizes linear transformations and change of basis operations via matricesto which he draws the analogy of translating text between English and Bulgarian, a nice analogy. He also emphasizes geometric Euclidean transformations as the prototype of linear transformationsrotations, dilations, projections, reflections.He covers all the usual topics in linear algebravector spaces, inner product spaces, linear independence, orthogonality, eigenspaces and diagonalization.etc. first before beginning a fairly complete discussion of the basics of differential equations that's somewhat more sophisticated then usual. As a result, he can use linear algebra throughout his discussion and that simplifies and clarifies the entire subject greatly without requiring heavy prerequisites: vector fields and continuous dynamical systems, complex eigenvalues, damped spring example, repeated eigenvalues, combination problems; stability, phaseplane analysis, analytic solutions. Linear differential operators and solutions to homogeneous and inhomogeneous linear differential equations, eigenfunctions, characteristic polynomial; kernel and image of a linear differential operator and phase space analysis. Despite how concise the notes are, they are very readable,careful and insightful.Also, there are some rather amusing cartoons at the homepage worth a chuckle. The notes may be too intensive for the average student,though, so warning is warranted.I highly recommend them for self study for strong
 students and instructors looking to challenge their students.
 Introduction to Differential Equations Patrick Dylan Zwick University of Utah Spring 2014 (PG/PG13) This is a standard course in elementary differential equations, pitched at a somewhat higher level then usual. It's prerequisites are plug and chug calculus and a good course in linear algebra.In other words, while students aren't expected to have rigorous mathematical training, they are expected to be able to understand and do simple proofs as one would expect in such elementary classes.While the website says the book for the course is Differential Equations and Linear Algebra by C.Henry Edwards and David E. PenneyI suspect the students in the course used the text relatively little. Zwick has written and posted a very extensive set of lecture notes, exercises and examscomplete with solutions,no less!for the course that more or less makes having an "official" text superfluous. The notes are extremely detailed with excellent examples and the author writes in a flowing, conversational and very friendly style that's going to make the notes enjoyable for students to read. More then that, the notes are often amusing and simultaneously insightful and most of the good stuff is in his wonderful footnotes at the bottom of the pages. .For a few examples, consider the footnote 5 on page 4 of Lecture 1,the title to the section in Lecture 2 introducing integration as a main method of solution," Integrate Your Way to Wealth and Happiness, or at Least a Solution" , and the very snarky footnote 1 on page 2 of Lecture 5. There are an enormous number of applications to the physical and social sciences with many pictorial aids, such as slope fields generated on computer. This is always a very good thing in elementary courses with students who may still fear mathematics, especially if it's not their thing. Notes like these will make undecided students consider making it their thing if they ace the course! In addition to these wonderful notes, Zwick has crafted an equally large number of additional exercises every bit as good as the textbook's, if not better.Amazingly, Zwick ends each lecture with commentary on the assigned problems in the texton the level of difficulty of a given problem and the concepts it's solution centers around. He encourages students to work the more difficult, ungraded assignments and gives some interesting digressions in order to motivate the students to do so. For example, he explains the brastichrome problem and it's relation to the calculus of variations in one of the end sections. As icing on the cake, he's posted a very large number of past exams from earlier iterations of the course with full solutionsfor students to work for practice.I have to say I was really surprised by the range, quality and quantity of Zwick's course materials. I can't think of a morestudent friendly resource for beginning students in differential equationsor any mathematical subject, for that matter. This site will be a remarkable resource for elementary differential equations for years to come.I suspect Zwick is beginning to draft his own textbook from these notes and exercises. I strongly encourage him to continue to make the materials available for free, even if he does eventually produce a published version. Either way, it will quickly become one of the standard sources from which to teach this course and I give it the highest possible recommendation for both students and teachers of this critical course.
 Ordinary and Partial Differential Equations Zachary S Tseng Penn State University Fall Semester 2013 Ordinary and Partial Differential Equations Zachary S Tseng Penn State University Sample Exams 20002014 (PG)An outstanding set of course materials for a year long first course in ordinary and partial
 differential equations that requires only single variable and multivariable pencil pushing calculus and some knowledge of matrix arithmetic. All the standard techniques are presented here clearly and in great detailseparation of variables, integrating factors, matrix methods,the Laplace transform, power series and Fourier series expansions, direction fields and much more. Not only is it very comprehensive and lucid, with many examples and clear presentation of both methods and concepts,but Tseng gives many exercises and exams with full solutions. And when I say he gives old exams with many solutions, I mean dozens of them for the student to work through going back 15 years at the second link! Tseng writes very clearly and instructivelythere's just enough hand holding for the beginner for them to be motivated to study this wonderful subject. There are many standard applications in the course of the presentation to both the physical and social sciences, such as Newton's law of cooling,mechanical and electrical vibrations, heat flow via Laplace's equation, the one dimensional oscillator and many more. One of the best study sources available online currently for such a course. Highly recommended as both a course text and study supplement. Contents 1 INTRODUCTION 2 FIRST ORDER DIFFERENTIAL EQUATIONS 3 SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 4. HIGHER ORDER LINEAR EQUATIONS 6. THE LAPLACE TRANSFORM 7. SYSTEMS OF TWO LINEAR DIFFERENTIAL EQUATIONS 9. NONLINEAR DIFFERENTIAL EQUATIONS AND STABILITY 10 PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES
 Differential Equations Hui Sun USCD Fall 2013 (PG)Good handwritten set of notes for an introductory differential equations course who's prerequisite was both basic calculus and linear algebra. The assumption of linear algebra as a prerequisite isn't really necessary for this kind of course,but I always thought it made things go a lot smoother and quicker if you don't have to introduce it on a need to know basis. The textbook for the course is an old favorite of mine since I first learned the subject out of an earlier edition, Elementary Differential Equations by William E. Boyce and Richard C. DiPrima. Contents: Ordinary differential equations: exact, separable, and linear; constant coefficients, undetermined coefficients, variations of parameters. Systems. Series solutions. Laplace transforms. Techniques for engineering sciences. Computing symbolic and graphical solutions using Matlab.Sun has good handwriting and a good choice of material and solved problems for the student. But it's clear he isn't writing these to supplant the textbook. As a result, they aren't really detailed enough to use as a course text by themselves.Recommended as a supplement.
 Differential Equations Gerhard Dangelmayr Colorado State University Fall Semester 2013 (PG)Large collection of"bullet point" style, supplementary notes to a standard plug and chug beginning course in differential equations. For reference, the required textbook for the course was Differential Equations by John Polking, Al Boggess and David Arnold. I've never seen this book, but I'm told by several friends who have taught from it it's quite good.( I do know it's insanely expensive, like most frequently assigned textbooks.) In any event, Danglmayer's notes Contents: First order equations Mathematical models Linear equations of second order The Laplace transform Linear systems of arbitrary order and matrices Nonlinear systems and phase plane analysis Numerical methods. The notes are clearly not intended to act as anything more then a supplementthe vast bulk of the notes are solved problems and computer generated graphs from the textbook problems. They're clear and do a nice job reviewing all the main points of the course and supplying a boatload of examplesbut they don't have enough systematic presentation of the concepts to be used as an actual textbook. What they can do is supply worked examples for either the assigned text or another cheap "main" textbook in introductory DEs that lacks such examples. For example, Nick Kovasar's notes paired with these could very well serve as a full textbook for such a basic course. Recommended as a supplement.
 Differential Equations Class Notes Walter Schreiner Christian Brothers University (Tennessee) (PG)A quite solid online text/ lecture notes set for an introductory differential equations course that requires basic calculus as a prerequisite.Contents:Chapter 1 The Wonderful World of Differential Equations Chapter 2 Separable Equations Chapter 3 Direction Fields And Solution Curves Chapter 4 Numerical Methods With First Order Linear Equations Chapter 5 First Order Linear Equations Chapter 6 Modeling With First Order Equations Chapter 8 Exact Equations Chapter 9 Second Order Linear Equations Chapter 10 The Laplace Transform Chapter 11 Solutions of Planar Linear Systems The notes are clear, focused and very well organized within each chapter. The notes also contain many pictures, concrete examples and the standard applications. Best of all, each "chapter" are supplemented by additional PDF files giving discussions about the MAPLE computer algebra system and how all these methods may be implemented via code. Such computer programming sidebars are very important for showing students how differential equations are used in the real world. Sadly, overall,he doesn't do such a good job of organizing the individual chaptersthe author places Chapter 4 and Chapter 5 in the reverse incorrect order and the chapters are misnumbered. These are easily rectified by the mathematically experienced,but will confuse the beginner. Therefore, all potential usersyou've been warned now! It's a good example of my initial warnings to the users of this website:Lecture notesno matter how well written they are and with nice pictures,etc.are NOT textbooks and are bound to contain not only errors of content, but of organization! Again,thoughthis is a simple file/chapter order screwup and now that it's been spotted, shouldn't be a problem.This minor defect aside, Schreiner has written a fine first course on differential equations and I'd heartily recommend it to both students and teachers of introductory differential equations.
 Differential Equations Nestor Guillen University of California at Los Angeles Spring 2012 (PG13) Very concise and careful set of lecture notes and exercises for a strong first course in differential equations. Contents: (1) The notion of a differential equation. (2) Basic tricks for nonlinear 1st order equations: methods to find explicit formulas for the solutions in several special cases (a) Separation of variables (b) Implicit integration. (3) The general theory of Linear differential equations (a) Integrating factor (b) Homogeneous and inhomogeneous equations (c) Method of undetermined coefficients (d) Variation of parameters (4) The existence and uniqueness theorems. Picard’s Theorem. (5) Autonomous equations and semigroups. (6) Phase diagram analysis (or: how to understand solutions in the absence of formulas) (7) Basic linear algebra and its application to linear systems of differential equations with constant coefficients. Concretely: eigenvalues and eigenvectors, exponential of a matrix, normal form, the Wronskian, fundamental system of solutions. There are a surprisingly large number of examples in notes that are quite concise and relatively brief. It's stated that only standard first courses in single and multivariable calculus and a basic course in linear algebra are needed as prerequisites for these notes. For the most part, that's truebut Guillen's notes are more mathematical and prooforiented then the average course of this nature. The proof of Picard's theorem, fixed point interation and the matrix exponential are topics that are present here one would think too difficult without an advanced calculus course. He recommends that the harder proofs be skipped in a basic course. My guess is that this course was a combined standard and honors course and the more theoretical sections were skipped by the "average" section of the course. In any event, Guillen writes very well and produces an excellent course for mathematics and physics majors. Recommended for strong students and teachers of differential equations.
 Ordinary Differential Equations with Linear Algebra Todd Kapitula Calvin University These are the notes that form an earlier draft of his currently online but password protected and available only to his students textbook on differential equations and linear algebra. They form the basis for a one year unified course in both elementary differential equations and basic linear algebra. If these notes are an indication, the upcoming book should become a standard textbook for such courses when it’s finally published. The book is careful, comprehensive and rigorous with very minimal prerequisites basically just a good but nonrigorous course in single variable calculus. A course like this is usually offered at more advanced levels, while assuming the basics of real variables. The author does not do this. Although his presentation is careful and all major results are proven, particularly in the sections on linear algebra, he keeps the course level low enough that no hard analysis is really needed. He reserves more sophisticated results and machinery in differential equationssuch as the general existence and uniqueness theorem and Picard iterationfor more advanced courses. Instead, he uses the tools of linear algebra to develop both the calculational and theoretical structure of the solution spaces of ODEs in ways that are usually reserved for more advanced coursessuch as the complex exponential and it’s role in solving ODEs and systems of ODEs in R^{n }. There are also many sophisticated applications, such as the use of spectral theory to model Markov processes. Computer algebra systems are also utilized to good effect. In short, a very strong course for mathematics and physics majors that demonstrates the power of unifying these 2 critical subjects. Highly recommended.