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Other titles in the Undergraduate Texts in Mathematics series:
A First Course in Differential Equations (Undergraduate Texts in Mathematics)by J. David Logan
Synopses & ReviewsPublisher Comments:This concise and uptodate textbook is designed for the standard sophomore course in differential equations. It treats the basic ideas, models, and solution methods in a user friendly format that is accessible to engineers, scientists, economists, and mathematics majors. It emphasizes analytical, graphical, and numerical techniques, and it provides the tools needed by students to continue to the next level in applying the methods to more advanced problems. There is a strong connection to applications with motivations in mechanics and heat transfer, circuits, biology, economics, chemical reactors, and other areas. Exceeding the first edition by over one hundred pages, this new edition has a large increase in the number of worked examples and practice exercises, and it continues to provide templates for MATLAB and Maple commands and codes that are useful in differential equations. Sample examination questions are included for students and instructors. Solutions of many of the exercises are contained in an appendix. Moreover, the text contains a new, elementary chapter on systems of differential equations, both linear and nonlinear, that introduces key ideas without matrix analysis. Two subsequent chapters treat systems in a more formal way. Briefly, the topics include: * Firstorder equations: separable, linear, autonomous, and bifurcation phenomena; * Secondorder linear homogeneous and nonhomogeneous equations; * Laplace transforms; and * Linear and nonlinear systems, and phase plane properties.
Synopsis:This concise, uptodate textbook is designed for the standard sophomore course in differential equations. The basic ideas, models, and solution methods are presented in a user friendly format that is accessible to engineers, scientists, economists, and mathematics majors.
About the AuthorJ. David Logan is Willa Cather Professor of Mathematics at the University of Nebraska Lincoln. His extensive research is in the areas of theoretical ecology, hydrogeology, combustion, mathematical physics, and partial differential equations. He is the author of six textbooks on applied mathematics and its applications, including Applied Partial Differential Equations, 2nd edition (Springer 2004) and Transport Modeling in Hydrogeochemical Systems (Springer 2001).
Table of ContentsPreface to the Second Edition. To the Student. 1. Differential Equations and Models. 1.1 Introduction. 1.2 General Terminology. 1.2.1 Geometrical Interpretation. 1.3 Pure Time Equations. 1.4 Mathematical Models. 1.4.1 Particle Dynamics. 1.5 Separation of Variables. 1.6 Autonomous Differential Equations. 1.7 Stability and Bifurcation 1.8 Reactors and Circuits. 1.8.1 Chemical Reactors. 1.8.2 Electrical Circuits 2. Linear Equations and Approximations. 2.1 FirstOrder Linear Equations. 2.2 Approximation of Solutions. 2.2.1 Picard Iteration*. 2.2.2 Numerical Methods. 2.2.3 Error Analysis. 3. SecondOrder Differential Equations. 3.1 Particle Mechanics 3.2 Linear Equations with Constant Coefficients. 3.3 The Nonhomogeneous Equation 3.3.1 Undetermined Coefficients. 3.3.2 Resonance. 3.4 Variable Coefficients. 3.4.1 CauchyEuler Equation. 3.4.2 Power Series Solutions*. 3.4.3 Reduction of Order*. 3.4.4 Variation of Parameters. 3.5 Boundary Value Problems and Heat Flow*. 3.6 HigherOrder Equations. 3.7 Summary and Review. 4. Laplace Transforms. 4.1 Definition and Basic Properties. 4.2 Initial Value Problems. 4.3 The Convolution Property. 4.4 Discontinuous Sources. 4.5 Point Sources. 4.6 Table of Laplace Transforms. 5. Systems of Differential Equations. 5.1 Linear Systems. 5.2 Nonlinear Models. 5.3 Applications. 5.3.1 The LotkaVolterra Model. 5.3.2 Models in Ecology. 5.3.3 An Epidemic Model. 5.4 Numerical Methods. 6. Linear Systems. 6.1 Linearization and Stability. 6.2 Matrices*. 6.3 TwoDimensional Linear Systems. 6.3.1 Solutions and Linear Orbits. 6.3.2 The Eigenvalue Problem. 6.3.3 Real Unequal Eigenvalues. 6.3.4 Complex Eigenvalues. 6.3.5 Real, Repeated Eigenvalues. 6.3.6 Stability. 6.4 Nonhomogeneous Systems*. 6.5 ThreeDimensional Systems*. 7. Nonlinear Systems. 7.1 Linearization Revisited. 7.1.1 Malaria*. 7.2 Periodic Solutions. 7.2.1 The Poincar´eBendixson Theorem. Appendix A. References. Appendix B. Computer Algebra Systems. B.1 Maple. B.2 MATLAB. Appendix C. Sample Examinations. D. Index.
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