Differential Equations with Applications and Historical Notes | Pure and Applied Mathematics Textbook | Perfect for University Students and Researchers in Engineering, Physics, and Applied Mathematics

There are three introductory differential equations textbooks which I find of lasting utility:(1) Hirsch and Smale, Differential Equations, Dynamical Systems and Linear Algebra (1974, Academic Press).(2) Hubbard and West, Differential Equations, A Dynamical Systems Approach (1991, Springer).(3) Simmons, Differential Equations with Applications and Historical Notes (1991, second edition).As an adjunct, one can hardly ignore Dieudonne's Infinitesimal Calculus (1971, chapter eleven, Hermann).Now, my first introductory course in differential equations occurred late 1996, where not one of the above mentioned texts was ever referenced. What a pity ! Now, regarding Simmons:(1) I begin with Chapter 13. Yes, I begin with chapter thirteen ! Read: "One of the main recurring themes of this book has been the idea that only a few simple types of differential equations can be solved explicitly in terms of known elementary functions." That line should be the first statement met by a new student to the subject. It is interesting that Simmons situated existence and uniqueness (chapter 13) ahead of numerics (chapter fourteen), whereas Hubbard and West accomplish the same in reverse (numerics chapter three, existence and uniqueness chapter four). Simmons shows "logical dependence" of chapters (page xxi), you will notice that thirteen and fourteen stand independent.(2) If you enjoy practicing problem-solving skills, this book is helpful. Thirty pages of solutions to those problems are part and parcel of the enterprise (although, temptation to look at the solution ahead of time should be admonished). Hints are provided (for instance, page 309, #3 d, "use the trigonometric expansion," or, page 479, #1: "write conditions...in the form...and use polar coordinates to deduce a contradiction."). Excellent pedagogy !(3) Appendices offer further historical, bibliographic information, but accomplish much more than just that. A highlight: Appendix B, proof of Lienard's theorem (pages 497-501). The proof here is as lucid as any other exposition (compare: Hartman, page 179, Ordinary Differential Equations). A nice detour, the discussion of Lagrange multipliers. Read: "This technique has two major features important in theoretical work, it does not disturb the symmetry of the problem and it removes the side condition." (page 516).Meet Euler (see Appendix A, pages 136-145), which segues into a discussion of continued fractions and graph theory.(4) You get practice with the computational (the usual assortment of separation of variables, laplace transforms, etc.), you get physical applications (Newton's laws plus Hermite polynomials and quantum mechanics,then Lagrange and Hamilton), finally you get some abstraction--but, not too abstract (see pages 282-284, those famous inequalities of Schwarz and Minkowski). In any event the mix of historical, applied, and theoretical engagingly espoused throughout.(5) A highlight: those special functions of mathematical physics, Legendre and Bessel, especially pages 348-380.A highlight, the survey of Nonlinear Equations (chapter 11): "without the aid of sophisticated mathematical machinery, requiring little more than elementary differential equations and two-dimensional vector algebra." (page 441). Complementary to this discussion, please study Sokolnikoff and Redheffer (1966, Mathematics of Physics and Modern Engineering, chapter two, nonlinear differential equations).(6) I can add little more to the extant reviews, as this is indeed a wonderful introductory textbook.I still remember reading that "College Mathematics Journal" special issue, devoted to Differential Equations(Volume 25, Number 5, November 1994). As I noted earlier, my first introductory course occurred 1996. Printed inside of this journal, names of board-of-editors for this issue. The name of the instructor for my 1996 course was included there. What a pity that the introductory course never referenced any one of my three favorite differential equations textbooks. Simmons, Hubbard & West, Hirsch & Smale-- are highly recommended for a first course.Do not forget to read Hubbard: What it Means to Understand a Differential Equation, in that CMJ issue !