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### Henry J. Ricardo

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2nd Edition - February 24, 2009

Author: Henry J. Ricardo

Language: EnglishHardback ISBN:

9 7 8 - 0 - 1 2 - 3 7 4 7 4 6 - 4

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 8 8 6 0 3 - 9

A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the… Read more

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*A Modern Introduction to Differential Equations, Second Edition,* provides an introduction to the basic concepts of differential equations.

The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equations and systems of differential equations; and systems of nonlinear equations. Each chapter concludes with a summary of the important concepts in the chapter. Figures and tables are provided within sections to help students visualize or summarize concepts. The book also includes examples and exercises drawn from biology, chemistry, and economics, as well as from traditional pure mathematics, physics, and engineering.

This book is designed for undergraduate students majoring in mathematics, the natural sciences, and engineering. However, students in economics, business, and the social sciences with the necessary background will also find the text useful.

- Student friendly readability- assessible to the average student
- Early introduction of qualitative and numerical methods
- Large number of exercises taken from biology, chemistry, economics, physics and engineering
- Exercises are labeled depending on difficulty/sophistication
- End of chapter summaries
- Group projects

Undergraduate students studying differential equations.

Chapter 1 Introduction to Differential EquationsIntroduction1.1 Basic Terminology1.2 Solutions of Differential Equations1.3 Initial-Value Problems and Boundary-Value ProblemsSummaryProject 1-1 Chapter 2 First-Order Differential Equations Introduction2.1 Separable Equations 2.2 Linear Equations2.3 Compartment Problems2.4 Slope Fields 2.5 Phase Lines and Phase Portraits 2.6 Equilibrium Points: Sinks, Sources, and Nodes 2.7 Bifurcations 2.8 Existence and Uniqueness of SolutionsSummaryProject 2-1Project 2-2Chapter 3 The Numerical Approximation of Solutions Introduction3.1 Euler’s Method3.2 The Improved Euler Method 3.3 More Sophisticated Numerical Methods: Runge-Kutta and OthersSummary Project 3-1 Chapter 4 Second- and Higher-Order EquationsIntroduction4.1 Homogeneous Second-Order Linear Equations with Constant Coefficients4.2 Nonhomogeneous Second-Order Linear Equations with Constant Coefficients4.3 The Method of Undetermined Coefficients4.4 Variation of Parameters 4.5 Higher-Order Linear Equations with Constant Coefficients4.6 Higher-Order Equations and Their Equivalent Systems4.7 The Qualitative Analysis of Autonomous Systems4.8 Spring-Mass Problems4.9 Existence and Uniqueness4.10 Numerical Solutions SummaryProject 4-1 Chapter 5 Systems of Linear Differential Equations Introduction 5.1 Systems and Matrices5.2 Two-Dimensional Systems of First-Order Linear Equations5.3 The Stability of Homogeneous Linear Systems: Unequal Real Eigenvalues5.4 The Stability of Homogeneous Linear Systems: Equal Real Eigenvalues5.5 The Stability of Homogeneous Linear Systems: Complex Eigenvalues5.6 Nonhomogeneous Systems5.7 Generalizations: The n × n Case (n ≥ 3) Summary Project 5-1 Project 5-2Chapter 6 The Laplace TransformIntroduction 6.1 The Laplace Transform of Some Important Functions6.2 The Inverse Transform and the Convolution 6.3 Transforms of Discontinuous Functions6.4 Transforms of Impulse Functions—The Dirac Delta Function6.5 Transforms of Systems of Linear Differential Equations6.6 A Qualitative Analysis via the Laplace TransformSummary Project 6-1Chapter 7 Systems of Nonlinear Differential EquationsIntroduction7.1 Equilibria of Nonlinear Systems7.2 Linear Approximation at Equilibrium Points7.3 The Poincaré-Lyapunov Theorem 7.4 Two Important Examples of Nonlinear Equations and Systems7.5 Van Der Pol’s Equation and Limit CyclesSummaryProject 7-1Appendix A Some Calculus Concepts and Results A.1 Local Linearity: The Tangent Line Approximation A.2 The Chain Rule A.3 The Taylor Polynomial / Taylor Series A.4 The Fundamental Theorem of Calculus (FTC) A.5 Partial Fractions A.6 Improper Integrals A.7 Functions of Several Variables / Partial Derivatives A.8 The Tangent Plane: The Taylor Expansion of F(x, y) Appendix B Vectors and Matrices B.1 Vectors and Vector Algebra; Polar Coordinates B.2 Matrices and Basic Matrix Algebra B.3 Linear Transformations and Matrix Multiplication B.4 Eigenvalues and Eigenvectors Appendix C Complex Numbers C.1 Complex Numbers: The Algebraic ViewC.2 Complex Numbers: The Geometric ViewC.3 The Quadratic Formula C.4 Euler’s FormulaAppendix D Series Solutions of Differential Equations D.1 Power Series Solutions of First-Order Equations D.2 Series Solutions of Second-Order Linear Equations: Ordinary Points D.3 Regular Singular Points: The Method of Frobenius D.4 The Point at Infinity D.5 Some Additional Special Differential EquationsAnswers/Hints to Odd-Numbered ExercisesIndex

- No. of pages: 536
- Language: English
- Edition: 2
- Published: February 24, 2009
- Imprint: Academic Press
- Hardback ISBN: 9780123747464
- eBook ISBN: 9780080886039

HR

Henry J. Ricardo works at Medgar Evers College of the City University of New York in Brooklyn, USA.

Affiliations and expertise

Medgar Evers College of the City University of New York, Brooklyn, USA