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Full Description
These materials, which have been developed and thoroughly class tested over a period of several years by the authors, are intended for use in courses in differential equations taught at the sophomore/junior level in American colleges and universities. Prerequisites for using these materials is the calculus of one variable, although calculus of several variables, and linear algrbra, are recommended. The text covers the standard topics in first and second order equations, power series solutions, first order systems, Laplace transforms, numerical methods and stability of non-linear systems. For historical completeness, the authors have also included the solution methods of Bernoulli, Clariaut, and Lagrange, if instructors wish to cover them. Liberal use is made of programs in Mathemtica, both for symbolic computations and graphical displays. The programs are described in separate sections, as well as in the accompanying Mathematica notebooks. The book has been designed so that it can be read with or without Mathematica and no pre-requisite knowledge of Mathematica is required to use the book, not to deal with the Mathematica programs contained in ODE.m, the special Mathematica software package residing on the CD-ROM accompanying the text. ODE.m enables students to solve differential equations, much as a calculator would. The CD-ROM, in addition to containing ODE.m, also contains the Mathematica solution of worked examples, the Mathematica solution of exercises, a protrait gallery of some 48 mathematicians who have contributed to the field of differential equations, a selction of various Mathematica noteboosk, Mathematica movies and sample labs for students. Mathematica programs and additional problem/example files will be made available online through the TELOS Web site (www.telospub.com) and the author's dedicated web site:
Contents
1. Basic Concepts.- 2. Using Mathematica.- 3. First-Order Differential Equations.- 4. The Package ODE • m.- 5. Existence and Uniqueness of Solutions of First-Order Differential Equations.- 6. Applications of First-Order Equations I.- 7. Applications of First-Order Equations II.- 8. Second-Order Linear Differential Equations.- 9. Second-Order Linear Differential Equations with Constant Coefficients.- 10. Using ODE to Solve Second-Order Linear Differential Equations.- 11.1 Mass-Spring Systems.- 12. Higher-Order Linear Differential Equations.- 13. Numerical Solutions of Differential Equations.- 14. The Laplace Transform.- 15. Systems of Linear Differential Equations.- 16. Phase Portraits of Linear Systems.- 17. Stability of Nonlinear Systems.- 18. Applications of Linear Systems.- 19. Applications of Nonlinear Systems.- 20 Power Series Solutions of Second-Order Equations.- 21. Frobenius Solutions of Second-Order Equations.- A. Appendix: Review of Linear Algebra and Matrix Theory.- A.l Vector and Matrix Notation.- A.2 Determinants and Inverses.- A.3 Systems of Linear Equations and Determinants.- A.4 Eigenvalues and Eigenvectors.- A.5 The Exponential of a Matrix.- A.6 Abstract Vector Spaces.- A.7 Vectors and Matrices with Mathematica.- A.8 Solving Equations with Mathematica.- A.9 Eigenvalues and Eigenvectors with Mathematica.- B. Appendix: Systems of Units.- Answers.- General Index.- Name Index.
