Differential Equations: (Miscellaneous/Catalogs S.)

This text on differential equations seeks to strike a balance between analytic, numeric and graphical techniques, emphasizes modelling, stresses qualitative theory throughout the course, uses technology, presents linear and nonlinear systems in parallel, and includes an introduction to discrete models of dynamical systems.

Table of Contents:
Part 1 First-order differential equations: modelling via differential equations; analytic technique - separation of variables; qualitative technique - slope fields; numerical technique - Euler's method; existence and uniqueness of solutions; equilibria and the phase line bifurcations; linear differential equations; changing variables. Part 2 First-order systems: the predator-prey model; systems of differential equations; graphical representation of solutions of systems; second-order equations and the harmonic oscillator; Euler's method for autonomous systems; qualitative analysis; the Lorenz equations. Part 3 Linear systems: properties of linear systems and the linearity principle; straight-line solutions; phase planes for linear systems with real Eigenvalues; complex Eigenvalues; the special cases - repeated and zero Eigenvalues; one-parameter families of linear systems; linear systems in three dimensions. Part 4 Nonlinear systems: equilibrium point analysis; Hamiltonian systems; dissipative systems; nonlinear systems in three dimensions. Part 5 Forcing and resonance: forced harmonic oscillators; sinusoidal forcing and resonance; qualitative analysis of the forced harmonic oscillator; the tacoma narrows bridge; periodic forcing of nonlinear systems and chaos. Part 6 Discrete dynamical systems: the discrete logistic equation; fixed and periodic runs; bifurcations; chaos; chaos in the Lorenz system. Part 7 Numerical methods: numerical error in Euler's method; improving Euler's method; the Runge-Kutta method; the effects of finite arithmetic. Part 8 Laplace transforms: Laplace transforms and second-order equations; impulse-forcing and delta functions; the qualitative theory of Laplace transforms;

Review :
"I liked the emphasis on qualitative and numerical solution methods in addition to the more traditional analytic techniques. Students come away from a course using this text with a better understanding of what an ordinary differential equation (ODE) is and what it means to "solve" such an equation." "I really liked the way that systems are introduced at the beginning of the text and stressed throughout the remainder of the text. I thoroughly enjoyed the systems approach to solving DE's and my students did, too. In fact, when we hit the "lucky guess" methods in later sections, my students found these traditional methods of solving DE's to be a snap compared to the algebra they used when solving systems. . . I liked the continuous emphasis on qualitative analysis. It was made evident in the first section of the text that graphical analysis would be used as a significant problem solving tool and the value of qualitative analysis was repeatedly stressed throughout the remainder of the text. . . I liked the balance between graphical analysis and the traditional symbolic analysis required in your typical DE course. I do not feel in the least that my students were shortchanged on symbolic manipulations required to be successful in later courses. On the other hand, the authors were careful not to go overboard and introduce one symbolic technique after another. I really feel they struck a fair balance." "It is clearly the best undergraduate ODE text that I have ever looked at, and I am pushing to have the text adopted for all our undergraduate differential courses. . . The authors have jettisoned the classic "cookbook" approach in favor of one which fosters critical thinking. The students need to formulate, solve and interpret the solutions to the problems. In order to do this effectively, the students are taught some fundamentals of modeling and are presented with an arsenal of tools, including qualitative, numeric, and analytic approaches for solving differential equations." "There are many texts that claim to be written in the spirit of the so-called calculus reform movement. In my opinion, the text under review achieves this goal better than any book that I have seen. This book presents at a very elementary level the kinds of material that opens the door for the understanding of many of the great achievements in dynamical systems that have been accomplished over the last 30 years."