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Introduction to differential equations. (English) Zbl 0624.34001

Englewood Cliffs, New Jersey: Prentice-Hall, Inc. X, 628 p.; £41.40 (1987).
This book is devoted exclusively to the practical part of elementary ordinary differential equations. To emphasize the author’s consistency in keeping with this line, note that he doesn’t give even the proof of existence of solutions of the Cauchy problem. However, there are many interesting examples of applications to mechanical problems, chemical kinetics, electromagnetic theory, elasticity, economy, biology, acoustic and quantum mechanics. These examples are sensible and well motivated. The book is draw up clearly, legibly and with great care. I am sure that it can be very useful for everybody who is interested in applications of differential equations, even readers whose mathematical knowledge is rather limited.
Here is the table of contents of the book. Chapter 2 - first order differential equations, method of separation of variables, exact equations, integrating factors, linear equations, various substitutions. Chapter 3 - second order equations, nonlinear second order equations that can be reduced to first order differential equations, second order linear equations with constant coefficients. Chapter 4 - linear equation of arbitrary order, general properties, the method of undeterminal coefficients, and variation of parameters. Chapter 5 - series solutions of differential equations, review of power series and analytic functions, linear equations with analytic coefficients, Hermite and Bessel’s equations, regular singular points. Chapter 6 - basic properties of Laplace transformation, using the Laplace transformation to solve initial value problems, linear nonhomogeneous differential equations with discontinuous nonhomogeneous terms, differential equations with impulsive forcing. Chapter 7 - Systems of linear equations, general properties matrices, homogeneous systems, the eigenvalue-eigenvector method of solutions, application of Laplace transformation. Chapter 8 - numerical methods, Euler’s method, Runge-Kutter method, systems of differential equations. Chapter 9 - qualitative analysis, analysis of scalar equation, linearization, Lyapunov stability. Chapter 10 - basic properties of Fourier series, convergence, integration and differentiation of Fourier series, some applications to ordinary differential equations. In the last chapter the heat equation and the wave equation are studied by using the method of separation of variables.
Reviewer: J.Myjak

MSC:

34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34A30 Linear ordinary differential equations and systems
34D20 Stability of solutions to ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
65J99 Numerical analysis in abstract spaces
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations