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Differential Equations: An Applied Approach
J. M. Cushing, University of Arizona

ISBN-10: 013044930X
ISBN-13: 9780130449306

Publisher: Prentice Hall
Copyright: 2004
Format: Cloth; 498 pp
Status: Out of Print

Suggested retail price: $128.00
This item is out of print and is no longer available for purchase.



For introductory courses in differential equations.

This modern introduction to differential equations covers traditional subjects, as well as modern topics such as fundamentals of dynamical systems theory and bifurcation theory. The volume emphasizes analyzing solutions rather than finding solution formulas, introduces numerical methods early in the text, and provides case studies for each subject area. Many applications are quite lengthy and detailed, and accompanied with real data and are drawn from many disciplines including many from biological subjects.

  • Most applied text: Cases studies with real data conclude each chapter.

    • There are 6 "themed" set of applications running throughout ends of chapters.

                    Population Dynamics: Sections 1.5.1, 3.6.1, 4.6.1, 7.5.1, 8.9.3, 9.7.2

                    Epidemics: Sections 2.6.1, 8.9.1, 9.7.2

                    Drug Kinetics: Sections 5.8.1, 6.5.1, 9.7.2

                    Objects in Motion: Sections 1.5.2, 3.6.2, 4.6.2, 5.8.2, 6.5.2, 7.5.2, 8.9.2

                    Oscillations: Sections 3.6.1, 5.8.2, 6.5.2, 8.9.3

                    Temperatures and Flows: Sections 2.6.2, 2.6.3, 3.6.3, 4.6.1

  • Traditional and modern topics covered—e.g. fundamentals of dynamical systems theory and bifurcation theory are extensively used, as well as traditional algebraic techniques.

    • Provide students with a introduction to traditional subjects and dynamical systems.

  • Solution approximation techniques—Ranging from graphical methods to analytic approximation methods.

    • Provides an introduction to perturbation methods, an important class of techniques that is rarely addressed in introductory courses.                 

  • Numerical methods using computers introduced.

    • Enables students to use numerical methods throughout the text.

  • Application modeling cycle emphasized, as organized around four fundamental steps—i.e. model derivation, model analysis, model interpretation and model refinement.

    • Enables students to see how applications are used to draw real conclusions and answer questions in scientific problems.

  • 1,700 exercises—Both drill problems and theoretical problems.

    • Provides students with a variety problems that extend material given or introduce closely related material.

  • The author is a well-known math biologist—His background helps in creating on usually applied book.

  • Hundreds of examples—Motivating examples, pedagogical offset examples, scientific offset examples.

    • Provides students with many examples that reinforce the text material.

(NOTE: Each chapter concludes with Chapter Summary & Exercises and Applications.)

1. First Order Equations.

The Fundamental Existence Theorem. Approximation of Solutions. Another Numerical Algorithm.



2. Linear First Order Equations.

The Solution of Linear Equations. Properties of Solutions. The Method of Undetermined Coefficients. Autonomous Linear Equations.



3. Nonlinear First Order Equations.

Autonomous Equations. Separable Equations. Change of Variables. Approximation Formulas.



4. Systems and Higher Order Equations.

Systems and Higher Order Equations. Approximating Solutions of Systems. Linear Systems of Equations.



5. Homogeneous Linear Systems and Higher Order Equations.

Introduction. Solving Homogeneous Systems. Homogeneous Second Order Equations. Phase Plane Portraits. Matrices and Eigenvalues. Higher Order Systems.



6. Nonhomogeneous Linear Systems.

Introduction. The Method of Undetermined Coefficients. The Variation of Constants Formula. Matrix Notation.



7. Approximations and Series Solutions.

Taylor Polynomials and Picard Iterates. A Perturbation Method. Power Series Solutions.



8. Nonlinear Systems.

Introduction. Equilibria. The Linearization Principle. Local Phase: Plane Portraits. Global Phase Plane Portraits. Bifurcations. Higher Dimensional Systems.



9. Laplace Transforms.

Introduction. The Laplace Transform. Linearity and the Inverse Laplace Transform. Properties of the Laplace Transform. Solution of Initial Value Problems.



10. Answers to Selected Exercises.

For Differential Equations


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