Differential Equations with Boundary Value Problems by Selwyn Hollis

Market:

Mathematics departments whose differential equations classes serve students in engineering, physical/biological sciences, and mathematics. Departments which have absorbed the best aspects of calculus "reform" without tossing out algebraic methods completely. Departments desiring an attractive alternative to reform books for a course for mathematics majors.

Primary talking points:

1. A modern treatment that strikes a balance between tradition and reform.
2. Exposition that is clear, concise, reader-friendly, and mathematically sound.
3. Flexibility. Can be used in courses that fall along a large portion of the continuum between the extremes of anti-technology and reform.

Fewer pages than Edwards & Penney, for example. Why?

Two reasons:

1. Power-series methods are distilled to two sections (the second of which emphasizes polynomial solutions and their orthogonality properties.)
2. The exposition is generally more concise, and technology-oriented ``projects" are not included. (Left for manuals.)

How is it different (and better) than Boyce & DiPrima?

• Less emphasis on categorization and solution techniques for first-order equations.
• Better applications of nonlinear first-order equations and systems.
• Deemphasis of power-series solutions.
• Deemphasis of linear equations of order higher than two.
• Numerical methods are integrated into the early chapters.
• Far more interesting treatment and applications of nonlinear systems.

Author's Suggested Highlights:

• Homogeneity (p. 16) is defined as a property of the set of solutions of an equation, not as whether a 0 appears in a certain "slot" in the equation.
• Section 2.2, p. 39. Mixing problems with variable volume.
• Problem 10, p.58. This problem illustrates the goal of connecting graphical/geometric ideas with the behavior of solutions.
• The equation y' = g(y/ t), p.61. Such equations are usually (and unfortunately) called ``homogeneous" equations. We do not use that terminology, and we treat the equation simply for what it is: an equation that can be made separable by an appropriate substitution.
• Problems 24--30, p. 68. These problems illustrate that Bernoulli equations are a special case of a broader class of equations that can be made linear by means of an appropriate substitution.
• Problems 15--18, p. 71. These problems use reduction of order to draw the connection between differential equations and conservation of energy.
• Section 3.5.1. Bernoulli and Riccati equations arise in nonlinear models of motion with resistance.
• Section 3.5.2, p. 81. Mixing problems in draining tanks.
• Section 3.5.3, p. 84. A nonlinear "surge-suppression" circuit with a "varistor." (Maple and Mathematica manuals will treat the diode in a half-wave rectifier circuit as a nonlinear resistor.)
• Section 4.2, p. 97. Emphasis on consequences of uniqueness. (A basis for the qualitative analysis in the following section.)
• Section 4.4. A thorough treatment of the logistic population model. Harvesting, curve-fitting, and seasonal coefficients are discussed.
• Section 4.5. A concise treatment of the Euler, Heun, and Runge-Kutta methods with careful discussion of error estimates.
• Section 4.6. A brief first look at systems, preceding the chapters on linear second-order equations.
• Section 5.2. 3D plots of solutions of second-order equations.
• Section 5.5. Computation of Green's functions by variation of constants.
• Sections 5.6-5.7. A distilled treatment of power-series solutions. The method of Frobenius is the subject of problems 27--31, p. 178. Section 5.7 emphasizes polynomial (i.e., terminating power-series) solutions of the Chebychev, Legendre, and Hermite equations.
• Chapter 6. (1.) Solutions are illustrated with both phase-plane graphs of y' versus y and graphs of y and y' versus t. (2) Exponential shift and complex solutions are used to simplify (and unify) computation of solutions.
• Section 7.6, p.256. An elegant connection between Green's functions and Laplace transforms is made via convolution.
• Sections 8.5-8.6. The term ``matrix exponential" and the associated notation are motivated by analogies with properties of the elementary exponential function. The matrix exponential is computed by means of both eigenpairs and, in Section 8.6, polynomial interpolation (via the Cayley-Hamilton theorem).
• Chapter 9. Very visually enhanced. Figure 17, p. 331, is a unique graphic that summarizes all of the possible types of phase portraits for linear systems in the plane.
• Chapter 10. No competing book contains this type of focused treatment of real applications of nonlinear systems. Of particular note is the use of the phase-plane to explain the behavior of the Fitzhugh-Nagumo nerve-conduction model. In section 10.5, models of mechanical systems such as the double pendulum and the three-body problem allow students to create fascinating computer animations.
• Chapter 11. Eigenvalue problems for Sturm-Liouville operators are presented in a way that is consistent with eigenvalue problems for matrices.

Quotes from John Davis, who used a draft in his class:

"The exposition here is much clearer and reader-friendly than in Boyce & DiPrima. Students who used early drafts of the manuscript reported that the balance between theory and examples was nice. They also commented on how the infusion of quality, relevant graphics from the start helped them develop a feel and appreciation of the qualitative aspects of differential equations. This proved especially valuable later in the course.

"Nonlinear equations are introduced early and often. The motivating first order applications are very diverse and not artificial as in most other texts. The usual population modeling and Newton mechanics are done, but so are sections on motion with nonlinear resistance, Torricelli's law, and nonlinear circuits!

"Unlike the traditional approach in Boyce & DiPrima which is heavy on series solutions, series solutions are deemphasized (but not abandoned as in Blanchard & Devaney!) which leaves more time to explore linear and nonlinear systems. The complex methods used throughout [Chapter 6] provide a uniform method for solving second-order constant-coefficient DEs. Numerical methods are also appropriately dealt with.

"The most glaring contrasts between Hollis and either Boyce & DiPrima or Blanchard & Devaney are the balance that is struck here between traditional and reform, the excellent exposition and use of graphics to reinforce qualitative concepts---without abandoning mechanics and the calculus--- and the robust (and often downright creative!) problem sets."

Quotes from reviewers:

"[The book] combines a classical selection of topics with a modern point of view, and as such is sure to be a welcome addition to the textbook literature on this subject."

"Hollis acknowledges and uses computer algebra and numerical simulation where appropriate but does not rely on them to the exclusion of good old-fashion algebra and calculus techniques, which will certainly be sharpened by users of this text. There seems to be a bit of a backlash against calculus reform these days, so perhaps analytic and algebraic techniques will be receiving renewed emphasis in our calculus courses for science majors. This means that the time is right for an ODE text like this one, and I predict that the Hollis text will be a hit."

"All theorems were stated with appropriate hypotheses, and when given, proofs were honest and correct. ... [Level] is probably the trickiest issue for an author of a differential equations book. The temptation, which Hollis admirably resists, is to slough over all the theory and just emphasize the various solution techniques. Instead, he strives for a middle ground where the mathematics is presented honestly and accurately---even if some of the harder proofs are omitted. ... Hollis takes great care to at least state the theorems accurately and give partial proofs where appropriate. So I think the level here is perfect. Students are taught the precise meaning of terms and hypotheses of the theorems. This is the right approach, and students who go on to higher-level math courses will benefit from it."

"I like the writing. It's mature, and not too informal or chatty. I think the author conveys a tone that this is a serious, deep and beautiful subject. The reader is treated with respect, not condescended to, as so many text do these days. I think my students would find this appealing. "

"Hollis makes good use of examples to motivate the development of the ideas and is good at pointing out recurring themes like linearity, superposition, eigenvalues, etc."

"The exercise sets (which are very thorough in range and difficulty) will suffice to cement the material in the minds of students."

"Overall, the text is well-written and quite clear, as well as interesting and pleasant to read. Its main strengths are the breadth of the covered material, plethora of examples and exercises, good pictures, and modern choice of topics."

"The strongest chapters in the book are 8 and 9. They are clear, contain few if any redundancies, and present interesting if a bit advanced examples. What is particularly nice is that there are ... many good illustrations and useful discussions."

"I really like the presentation of mechanical oscillations via the complexification of the force. This simplifies the calculations enormously."

"Some of the topics that present greatest difficulty to the students in my sophomore course are word problems [on] mechanical and electrical oscillations. This book has a much more detailed discussion of both than my current textbook."

"...the examples are very clear and helpful, and there are a great number of them. The exercises are good, and useful as well, ranging from easy to very hard and theoretical."

"Chapter 8, "Geometry of Autonomous Systems in the Plane" ... is a very nice and needed addition. In fact, it is the nicest chapter in the book."

"Overall, the book is very well organized, with a lot of thought given to the sequence of topics."

"The text is accurate. I have not encountered any incorrectness in the theory."

"The writing style is very nice. ... This book probably pays more attention to the style of explanation than all the other books the we have been or are using."

"The illustrations in this book (and there are many) are of very high quality ... [and] better than in most books, including the text that we are presently using."

"I strongly agree [with] and like the laid-back approach of the author to "technology," meaning CAS. They constitute a very useful tool; they might have even changed the way we approach math in the classroom, but it is wrong to elevate them to the level of the subject matter of the course itself."

"This text is up-to-date in the sense that it uses technology to illustrate many of the examples, and it's easy imagine supplementing the examples wth a CAS or other software."