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by Dennis G. Zill and Michael R. Cullen

Edition: 4TH 97Copyright: 1997

Publisher: Brooks/Cole Publishing Co.

Published: 1997

International: No

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This Fourth Edition of the expanded version of Zill's best - selling A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS places an even greater emphasis on modeling and the use of technology in problem solving and now features more everyday applications. Both Zill texts are identical through the first nine chapters, but this version includes six additional chapters that provide in - depth coverage of boundary - value problem - solving and partial differential equations, subjects just introduced in the shorter text. Previous editions of these two texts have enjoyed such great success in part because the authors pique students' interest with special features and in - text aids. Pre - publication reviewers also praise the authors' accessible writing style and the text's organization, which makes it easy to teach from and easy for students to understand and use. Understandable, step - by - step solutions are provided for every example. And this edition makes an even greater effort to show students how the mathematical concepts have relevant, everyday applications.

Among the boundary - value related topics covered in this expanded text are : plane autonomous systems and stability; orthogonal functions; Fourier series; the Laplace transform; and elliptic, parabolic, and hyperparabolic partial differential equations, and their applications.

**1. INTRODUCTION TO DIFFERENTIAL EQUATIONS.**

Definitions and Terminology.

Initial-Value Problems.

Differential Equations as Mathematical Models.

**2. FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Separable Variables.

Exact Equations.

Linear Equations.

Solutions by Substitutions.

**3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Linear Equations.

Nonlinear Equations.

Systems of Linear and Nonlinear Equations.

**4. DIFFERENTIAL EQUATIONS OF HIGHER ORDER.**

Preliminary Theory : Linear Equations.

Reduction of Order.

Homogeneous Linear Equations with Constant Coefficients.

Undetermined Coefficients - Superposition Approach.

Undetermined Coefficients - Annihilator Approach.

Variation of Parameters.

Cauchy-Euler Equations.

Systems of Linear Equations.

Nonlinear Equations.

**5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS.**

Linear Equations : Initial-Value Problems.

Linear Equations : Boundary-Value.

Problems.

Nonlinear Equations.

**6. SERIES SOLUTIONS OF LINEAR EQUATIONS.**

Review of Power Series.

Power Series Solutions.

Solutions about Ordinary Points.

Solutions about Singular Points.

Two Special Equations.

**7. LAPLACE TRANSFORM.**

Definition of the Laplace Transform.

Inverse Transform.

Translation Theorems and Derivatives of a Transform.

Transforms of Derivatives, Integrals and Periodic Functions.

Applications.

Dirac Delta Function.

Systems of Linear Equations.

**8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Preliminary Theory.

Homogeneous Linear Systems with Constant Coefficients.

Variation of Parameters.

Matrix Exponential.

**9. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS.**

Direction Fields.

Euler Methods.

Runge-Kutta Methods.

Multistep Methods.

Higher-Order Equations and Systems.

Second-Order Boundary-Value Problems.

**10. PLANE AUTONOMOUS SYSTEMS AND STABILITY.**

Autonomous Systems, Critical Points, and Periodic Solutions.

Stability of Linear Systems.

Linearization and Local Stability.

Modeling Using Autonomous Systems.

**11. ORTHOGONAL FUNCTIONS AND FOURIER SERIES.**

Orthogonal Functions.

Fourier Series.

Fourier Cosine and Sine Series.

Sturm-Liouville Problem.

Bessel and Legendre Series.

**12. PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES.**

Separable Partial Differential Equations.

Classical Equations and Boundary-Value Problems.

Heat Equation.

Wave Equation.

Laplace's Equation.

Nonhomogeneous Equations and Boundary Conditions.

Use of Generalized Fourier Series.

Boundary-Value Problems Involving Fourier Series in Two Variables.

**13. BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS.**

Problems involving Laplace's Equation in Polar Coordinates.

Problems in Polar and Cylindrical Coordinates : Bessel Functions.

Problems in Spherical Coordinates : Legendre Polynomials.

**14. INTEGRAL TRANSFORM METHOD.**

Error Function.

Applications of the Laplace Transform.

Fourier Integral.

Fourier Transforms.

**15. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS.**

Elliptic Equations.

Parabolic Equations.

Hyperbolic Equations.

Appendix I : Gamma Function.

Appendix II : Introduction to Matrices.

Appendix III : Laplace Transforms.

Answers to Odd-Numbered Problems.

Index.

Summary

This Fourth Edition of the expanded version of Zill's best - selling A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS places an even greater emphasis on modeling and the use of technology in problem solving and now features more everyday applications. Both Zill texts are identical through the first nine chapters, but this version includes six additional chapters that provide in - depth coverage of boundary - value problem - solving and partial differential equations, subjects just introduced in the shorter text. Previous editions of these two texts have enjoyed such great success in part because the authors pique students' interest with special features and in - text aids. Pre - publication reviewers also praise the authors' accessible writing style and the text's organization, which makes it easy to teach from and easy for students to understand and use. Understandable, step - by - step solutions are provided for every example. And this edition makes an even greater effort to show students how the mathematical concepts have relevant, everyday applications.

Among the boundary - value related topics covered in this expanded text are : plane autonomous systems and stability; orthogonal functions; Fourier series; the Laplace transform; and elliptic, parabolic, and hyperparabolic partial differential equations, and their applications.

Table of Contents

**1. INTRODUCTION TO DIFFERENTIAL EQUATIONS.**

Definitions and Terminology.

Initial-Value Problems.

Differential Equations as Mathematical Models.

**2. FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Separable Variables.

Exact Equations.

Linear Equations.

Solutions by Substitutions.

**3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Linear Equations.

Nonlinear Equations.

Systems of Linear and Nonlinear Equations.

**4. DIFFERENTIAL EQUATIONS OF HIGHER ORDER.**

Preliminary Theory : Linear Equations.

Reduction of Order.

Homogeneous Linear Equations with Constant Coefficients.

Undetermined Coefficients - Superposition Approach.

Undetermined Coefficients - Annihilator Approach.

Variation of Parameters.

Cauchy-Euler Equations.

Systems of Linear Equations.

Nonlinear Equations.

**5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS.**

Linear Equations : Initial-Value Problems.

Linear Equations : Boundary-Value.

Problems.

Nonlinear Equations.

**6. SERIES SOLUTIONS OF LINEAR EQUATIONS.**

Review of Power Series.

Power Series Solutions.

Solutions about Ordinary Points.

Solutions about Singular Points.

Two Special Equations.

**7. LAPLACE TRANSFORM.**

Definition of the Laplace Transform.

Inverse Transform.

Translation Theorems and Derivatives of a Transform.

Transforms of Derivatives, Integrals and Periodic Functions.

Applications.

Dirac Delta Function.

Systems of Linear Equations.

**8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Preliminary Theory.

Homogeneous Linear Systems with Constant Coefficients.

Variation of Parameters.

Matrix Exponential.

**9. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS.**

Direction Fields.

Euler Methods.

Runge-Kutta Methods.

Multistep Methods.

Higher-Order Equations and Systems.

Second-Order Boundary-Value Problems.

**10. PLANE AUTONOMOUS SYSTEMS AND STABILITY.**

Autonomous Systems, Critical Points, and Periodic Solutions.

Stability of Linear Systems.

Linearization and Local Stability.

Modeling Using Autonomous Systems.

**11. ORTHOGONAL FUNCTIONS AND FOURIER SERIES.**

Orthogonal Functions.

Fourier Series.

Fourier Cosine and Sine Series.

Sturm-Liouville Problem.

Bessel and Legendre Series.

**12. PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES.**

Separable Partial Differential Equations.

Classical Equations and Boundary-Value Problems.

Heat Equation.

Wave Equation.

Laplace's Equation.

Nonhomogeneous Equations and Boundary Conditions.

Use of Generalized Fourier Series.

Boundary-Value Problems Involving Fourier Series in Two Variables.

**13. BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS.**

Problems involving Laplace's Equation in Polar Coordinates.

Problems in Polar and Cylindrical Coordinates : Bessel Functions.

Problems in Spherical Coordinates : Legendre Polynomials.

**14. INTEGRAL TRANSFORM METHOD.**

Error Function.

Applications of the Laplace Transform.

Fourier Integral.

Fourier Transforms.

**15. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS.**

Elliptic Equations.

Parabolic Equations.

Hyperbolic Equations.

Appendix I : Gamma Function.

Appendix II : Introduction to Matrices.

Appendix III : Laplace Transforms.

Answers to Odd-Numbered Problems.

Index.

Publisher Info

Publisher: Brooks/Cole Publishing Co.

Published: 1997

International: No

Published: 1997

International: No

This Fourth Edition of the expanded version of Zill's best - selling A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS places an even greater emphasis on modeling and the use of technology in problem solving and now features more everyday applications. Both Zill texts are identical through the first nine chapters, but this version includes six additional chapters that provide in - depth coverage of boundary - value problem - solving and partial differential equations, subjects just introduced in the shorter text. Previous editions of these two texts have enjoyed such great success in part because the authors pique students' interest with special features and in - text aids. Pre - publication reviewers also praise the authors' accessible writing style and the text's organization, which makes it easy to teach from and easy for students to understand and use. Understandable, step - by - step solutions are provided for every example. And this edition makes an even greater effort to show students how the mathematical concepts have relevant, everyday applications.

Among the boundary - value related topics covered in this expanded text are : plane autonomous systems and stability; orthogonal functions; Fourier series; the Laplace transform; and elliptic, parabolic, and hyperparabolic partial differential equations, and their applications.

**1. INTRODUCTION TO DIFFERENTIAL EQUATIONS.**

Definitions and Terminology.

Initial-Value Problems.

Differential Equations as Mathematical Models.

**2. FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Separable Variables.

Exact Equations.

Linear Equations.

Solutions by Substitutions.

**3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Linear Equations.

Nonlinear Equations.

Systems of Linear and Nonlinear Equations.

**4. DIFFERENTIAL EQUATIONS OF HIGHER ORDER.**

Preliminary Theory : Linear Equations.

Reduction of Order.

Homogeneous Linear Equations with Constant Coefficients.

Undetermined Coefficients - Superposition Approach.

Undetermined Coefficients - Annihilator Approach.

Variation of Parameters.

Cauchy-Euler Equations.

Systems of Linear Equations.

Nonlinear Equations.

**5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS.**

Linear Equations : Initial-Value Problems.

Linear Equations : Boundary-Value.

Problems.

Nonlinear Equations.

**6. SERIES SOLUTIONS OF LINEAR EQUATIONS.**

Review of Power Series.

Power Series Solutions.

Solutions about Ordinary Points.

Solutions about Singular Points.

Two Special Equations.

**7. LAPLACE TRANSFORM.**

Definition of the Laplace Transform.

Inverse Transform.

Translation Theorems and Derivatives of a Transform.

Transforms of Derivatives, Integrals and Periodic Functions.

Applications.

Dirac Delta Function.

Systems of Linear Equations.

**8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.**

Preliminary Theory.

Homogeneous Linear Systems with Constant Coefficients.

Variation of Parameters.

Matrix Exponential.

**9. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS.**

Direction Fields.

Euler Methods.

Runge-Kutta Methods.

Multistep Methods.

Higher-Order Equations and Systems.

Second-Order Boundary-Value Problems.

**10. PLANE AUTONOMOUS SYSTEMS AND STABILITY.**

Autonomous Systems, Critical Points, and Periodic Solutions.

Stability of Linear Systems.

Linearization and Local Stability.

Modeling Using Autonomous Systems.

**11. ORTHOGONAL FUNCTIONS AND FOURIER SERIES.**

Orthogonal Functions.

Fourier Series.

Fourier Cosine and Sine Series.

Sturm-Liouville Problem.

Bessel and Legendre Series.

**12. PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES.**

Separable Partial Differential Equations.

Classical Equations and Boundary-Value Problems.

Heat Equation.

Wave Equation.

Laplace's Equation.

Nonhomogeneous Equations and Boundary Conditions.

Use of Generalized Fourier Series.

Boundary-Value Problems Involving Fourier Series in Two Variables.

**13. BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS.**

Problems involving Laplace's Equation in Polar Coordinates.

Problems in Polar and Cylindrical Coordinates : Bessel Functions.

Problems in Spherical Coordinates : Legendre Polynomials.

**14. INTEGRAL TRANSFORM METHOD.**

Error Function.

Applications of the Laplace Transform.

Fourier Integral.

Fourier Transforms.

**15. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS.**

Elliptic Equations.

Parabolic Equations.

Hyperbolic Equations.

Appendix I : Gamma Function.

Appendix II : Introduction to Matrices.

Appendix III : Laplace Transforms.

Answers to Odd-Numbered Problems.

Index.