This book is an interesting look at the modern theory of ordinary differential equations. It is arranged so that it requires little beyond calculus. On the other hand, it is not a traditional introduction to the subject: it only covers a few solution methods and there are no applications. This is also not a proofs book, although it does include proofs; it tries to illustrate the many ideas used in the subject.
I think the book tries to cover too many topics. It starts a lot of topics but doesn't have space to go into much depth, making it more of a survey than a true beginner's book. It does manage to pack a lot of material into a relatively small space. Roughly a third of the space is devoted to problems and hints. Most of the problems illustrate the material by asking the student to solve concrete differential equations (with numerical coefficients) and show that the solutions indeed have the properties claimed for the general case.
This is not a how-to book, although it teaches some of the simpler techniques. If you are looking for a book on how to solve differential equations, I recommend Bellman & Cook's Modern Elementary Differential Equations or the differential equations sections of Kreyszig's Advanced Engineering Mathematics.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.
Preface.- Introduction.- Historical Notes.- Exact Equations.- Elementary First Order Equations.- First Order Linear Equations.- Second Order Linear Equations.- Preliminaries to Existence and Uniqueness of Solutions.- Picard’s Method of Successive Approximations.- Existence Theorems.- Uniqueness Theorems.- Differential Inequalities.- Continuous Dependence on Initial Conditions.- Preliminary Results from Algebra and Analysis.- Preliminary Results from Algebra and Analysis (Contd.).- Existence and Uniqueness of Solutions of Systems.- Existence and Uniqueness of Solutions of Systems (Contd.).- General Properties of Linear Systems.- Fundamental Matrix Solution.- Systems with Constant Coefficients.- Periodic Linear Systems.- Asymptotic Behavior of Solutions of Linear Systems.- Asymptotic Behavior of Solutions of Linear Systems (Contd.).- Preliminaries to Stability of Solutions.- Stability of Quasi–Linear Systems.- Two–Dimensional Autonomous Systems.- Two–Dimensional Autonomous Systems (Contd.).- Limit Cycles and Periodic Solutions.- Lyapunov’s Direct Method for Autonomous Systems.- Lyapunov’s Direct Method for Non–Autonomous Systems.- Higher Order Exact and Adjoint Equations.- Oscillatory Equations.- Linear Boundary Value Problems.- Green’s Functions.- Degenerate Linear Boundary Value Problems.- Maximum Principles.- Sturm–Liouville Problems.- Sturm–Liouville Problems (Contd.).- Eigenfunction Expansions.- Eigenfunction Expansions (Contd.).- Nonlinear Boundary Value Problems.- Nonlinear Boundary Value Problems (Contd.).- Topics for Further Studies.- References.- Index