Long ago when the world was young, I discovered Michael Spivak’s Calculus book. I was entranced by it and was particularly fascinated by a section at the very end: “Suggested Reading.” (As you can see, I haven’t changed all that much.) I ended up reading many of the books suggested there; Hurewicz’s Lectures on Ordinary Differential Equations was one of them.
My memory is that (in the edition I had, which must have been an early one) Spivak said that while there were many elementary introductions to differential equations out there, most of them weren’t very good. Hurewicz, on the other hand, was more advanced but “widely admired.” I was a rank beginner at the time, and reading Hurewicz’s lectures did me very little good. Returning to it after all these years, I can see why, but I can also see the reason for the admiration.
The book is really an introduction to the theory of differential equations, aimed at readers who have a good bit of analysis under their belts. There are \(\varepsilon\)-\(\delta\) arguments throughout. In the first chapter Hurewicz proves existence and uniqueness of solutions (pages 7 to 10), continuous (page 12) and differentiable dependence (proved for systems of differential equations on page 29) on initial conditions, and so on. So this is not a beginner’s book.
For mature readers, however, it is a wonder. In a few pages, efficiently and cleanly, all of the hard theorems are proved. I particularly enjoyed Hurewicz’s proof of the existence and uniqueness of solutions. Rather than giving the usual proof via the Banach fixed-point theorem, he starts from Euler’s method for finding approximate solutions and proves that the approximations it produces in fact converge to a solution. It’s a two-for-one approach that simultaneously shows existence and provides a method for approximation. Very nice. Of course, being the thorough lecturer he is, he also gives the usual proof and later uses it to find the solution of autonomous linear systems of differential equations.
One thing that stands out as one reads is that in 1943, when these lectures were given, and even in 1956, when they were first published, linear algebra was not yet the common possession of all mathematicians. Notions such as dimension are not used, and linear dependence and independence are defined. One sees sentences like this:
If \(X_1,X_2,\dots, X_n\) is a linearly independent set of \(n\) vectors, then any vector can be written as a linear combination of the \(X\)’s; but this is true for no set of \(n<k\) vectors.
Eigenvalues and the characteristic polynomial are defined as necessary. So one concludes that at this point people who could understand uniform continuity and equicontinuous families and knew the Jordan curve theorem still needed explanations and definitions for what we now consider elementary linear algebra.
Some things that are central today are obviously missing here. There is no bifurcation theory and no discussion of deterministic chaos, for example. Chapters 4 and 5 are entirely about two-dimensional flows, in fact. So Hurewicz will not suffice today. But that is to be expected from lectures given 70 years ago.
Nevertheless, this is still a useful book. It provides quick access to real proofs of things that are usually just stated (if that!) in introductory books. For any student who has taken the introductory course and is mature enough to ask the “how do we prove?” question, it is hard to imagine a better choice of “suggested reading.”
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.
|Witold Hurewicz In Memoiam|
|Chapter 1||Differential Equations of the First Order in One Unknown Function|
|PART A THE CAUCHY-EULER APPROXIMATION METHOD|
|Section 1||Definitions. Direction Fields|
|2||Approximate Solutions of the Differential Equation|
|3||The Fundamental Inequality|
|4||Uniqueness and Existence Theorems Appendix to §4. Extension of the Existence Theorem|
|5||Solutions Containing Parameters|
|PART B CONTINUATION OF SOLUTIONS. OTHER METHODS|
|6||Continuation of Solutions|
|7||Other Methods of Solution|
|Chapter 2||Systems of Differential Equations|
|Section 1||Introduction. Vector Notation|
|3||The Lipschitz Conditions|
|4||The Fundamental Inequality|
|5||Existence and Properties of Solutions of the System|
|6||Systems of Higher Order|
|Chapter 3||Linear Systems of Differential Equations|
|PART A THE GENERAL LINEAR SYSTEM|
|Section 1||Introduction. Matrix Notation|
|2||Linear Dependence. Fundamental Systems|
|3||Solutions Expressed in Matrix Form|
|4||Reduction of Order of a System|
|PART B LINEAR EQUATIONS OF HIGHER ORDER|
|7||The Wronsky Determinant|
|8||Further Properties of the Fundamental System|
|9||Reduction of Order|
|10||The Nonhomogeneous Case|
|PART C LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS|
|12||Introduction. Complex Solutions|
|13||Characteristic Values and Vectors of a Matrix|
|14||Vectors Associated with Characteristic Values of a Matrix|
|15||The Solution in the Simplest Case|
|16||The Solution in the General Case|
|17||Homogeneous Equation of nth Order|
|Chapter 4||Singularities of an Autonomous System|
|PART A INTRODUCTION|
|Section 1||Characteristic Curves|
|PART B SINGULARITIES OF A LINEAR SYSTEM|
|6||Degenerate Nodal Points|
|9||Conclusion. Dynamical Interpretation|
|PART C NONLINEAR SYSTEMS|
|Chapter 5||Solutions of an Autonomous System in the large|
|4||Theorem of Poincare-Bendixson|
|6||Orbital Stability of Limit Cycles|
|7||Index of Simple Singularities|