# Differential Equations and Linear Algebra: Pearson New International Edition VitalSource eText (3e)

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For courses in Differential Equations and Linear Algebra.

Acclaimed authors Edwards and Penney combine core topics in elementary differential equations with those concepts and methods of elementary linear algebra needed for a contemporary combined introduction to differential equations and linear algebra. Known for its real-world applications and its blend of algebraic and geometric approaches, this text discusses mathematical modeling of real-world phenomena, with a fresh new computational and qualitative flavor evident throughout in figures, examples, problems, and applications. In the Third Edition, new graphics and narrative have been added as neededãyet the proven chapter and section structure remains unchanged, so that class notes and syllabi will not require revision for the new edition.

##### Table of contents

1.1 Differential Equations and Mathematical Models

1.2 Integrals as General and Particular Solutions

1.3 Slope Fields and Solution Curves

1.4 Separable Equations and Applications

1.5 Linear First-Order Equations

1.6 Substitution Methods and Exact Equations

CHAPTER 2. Mathematical Models and Numerical Methods

2.1 Population Models

2.2 Equilibrium Solutions and Stability

2.3 Acceleration–Velocity Models

2.4 Numerical Approximation: Euler's Method

2.5 A Closer Look at the Euler Method

2.6 The Runge–Kutta Method

CHAPTER 3. Linear Systems and Matrices

3.1 Introduction to Linear Systems

3.2 Matrices and Gaussian Elimination

3.3 Reduced Row-Echelon Matrices

3.4 Matrix Operations

3.5 Inverses of Matrices

3.6 Determinants

3.7 Linear Equations and Curve Fitting

CHAPTER 4. Vector Spaces

4.1 The Vector Space R3

4.2 The Vector Space Rn and Subspaces

4.3 Linear Combinations and Independence of Vectors

4.4 Bases and Dimension for Vector Spaces

4.5 Row and Column Spaces

4.6 Orthogonal Vectors in Rn

4.7 General Vector Spaces

CHAPTER 5. Higher-Order Linear Differential Equations

5.1 Introduction: Second-Order Linear Equations

5.2 General Solutions of Linear Equations

5.3 Homogeneous Equations with Constant Coefficients

5.4 Mechanical Vibrations

5.5 Nonhomogeneous Equations and Undetermined Coefficients

5.6 Forced Oscillations and Resonance

CHAPTER 6. Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

6.2 Diagonalization of Matrices

6.3 Applications Involving Powers of Matrices

CHAPTER 7. Linear Systems of Differential Equations

7.1 First-Order Systems and Applications

7.2 Matrices and Linear Systems

7.3 The Eigenvalue Method for Linear Systems

7.4 Second-Order Systems and Mechanical Applications

7.5 Multiple Eigenvalue Solutions

7.6 Numerical Methods for Systems

CHAPTER 8. Matrix Exponential Methods

8.1 Matrix Exponentials and Linear Systems

8.2 Nonhomogeneous Linear Systems

8.3 Spectral Decomposition Methods

CHAPTER 9. Nonlinear Systems and Phenomena

9.1 Stability and the Phase Plane

9.2 Linear and Almost Linear Systems

9.3 Ecological Models: Predators and Competitors

9.4 Nonlinear Mechanical Systems

CHAPTER 10. Laplace Transform Methods

10.1 Laplace Transforms and Inverse Transforms

10.2 Transformation of Initial Value Problems

10.3 Translation and Partial Fractions

10.4 Derivatives, Integrals, and Products of Transforms

10.5 Periodic and Piecewise Continuous Input Functions

CHAPTER 11. Power Series Methods

11.1 Introduction and Review of Power Series

11.2 Power Series Solutions

11.3 Frobenius Series Solutions

11.4 Bessel Functions

References for Further Study

Appendix A: Existence and Uniqueness of Solutions

Appendix B: Theory of Determinants

Answers to Selected Problems

Index

APPLICATION MODULES

The modules listed here follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook.

1.3 Computer-Generated Slope Fields and Solution Curves

1.4 The Logistic Equation

1.5 Indoor Temperature Oscillations

1.6 Computer Algebra Solutions

2.1 Logistic Modeling of Population Data

2.3 Rocket Propulsion

2.4 Implementing Euler's Method

2.5 Improved Euler Implementation

2.6 Runge-Kutta Implementation

3.2 Automated Row Operations

3.3 Automated Row Reduction

3.5 Automated Solution of Linear Systems

5.1 Plotting Second-Order Solution Families

5.2 Plotting Third-Order Solution Families

5.3 Approximate Solutions of Linear Equations

5.5 Automated Variation of Parameters

5.6 Forced Vibrations and Resonance

7.1 Gravitation and Kepler's Laws of Planetary Motion

7.3 Automatic Calculation of Eigenvalues and Eigenvectors

7.4 Earthquake-Induced Vibrations of Multistory Buildings

7.5 Defective Eigenvalues and Generalized Eigenvectors

7.6 Comets and Spacecraft

8.1 Automated Matrix Exponential Solutions

8.2 Automated Variation of Parameters

9.1 Phase Portraits and First-Order Equations

9.2 Phase Portraits of Almost Linear Systems

9.3 Your Own Wildlife Conservation Preserve

9.4 The Rayleigh and van der Pol Equations

CHAPTER 1. First-Order Differential Equations

1.1 Differential Equations and Mathematical Models

1.2 Integrals as General and Particular Solutions

1.3 Slope Fields and Solution Curves

1.4 Separable Equations and Applications

1.5 Linear First-Order Equations

1.6 Substitution Methods and Exact Equations

CHAPTER 2. Mathematical Models and Numerical Methods

2.1 Population Models

2.2 Equilibrium Solutions and Stability

2.3 Acceleration–Velocity Models

2.4 Numerical Approximation: Euler's Method

2.5 A Closer Look at the Euler Method

2.6 The Runge–Kutta Method

CHAPTER 3. Linear Systems and Matrices

3.1 Introduction to Linear Systems

3.2 Matrices and Gaussian Elimination

3.3 Reduced Row-Echelon Matrices

3.4 Matrix Operations

3.5 Inverses of Matrices

3.6 Determinants

3.7 Linear Equations and Curve Fitting

CHAPTER 4. Vector Spaces

4.1 The Vector Space R3

4.2 The Vector Space Rn and Subspaces

4.3 Linear Combinations and Independence of Vectors

4.4 Bases and Dimension for Vector Spaces

4.5 Row and Column Spaces

4.6 Orthogonal Vectors in Rn

4.7 General Vector Spaces

CHAPTER 5. Higher-Order Linear Differential Equations

5.1 Introduction: Second-Order Linear Equations

5.2 General Solutions of Linear Equations

5.3 Homogeneous Equations with Constant Coefficients

5.4 Mechanical Vibrations

5.5 Nonhomogeneous Equations and Undetermined Coefficients

5.6 Forced Oscillations and Resonance

CHAPTER 6. Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

6.2 Diagonalization of Matrices

6.3 Applications Involving Powers of Matrices

CHAPTER 7. Linear Systems of Differential Equations

7.1 First-Order Systems and Applications

7.2 Matrices and Linear Systems

7.3 The Eigenvalue Method for Linear Systems

7.4 Second-Order Systems and Mechanical Applications

7.5 Multiple Eigenvalue Solutions

7.6 Numerical Methods for Systems

CHAPTER 8. Matrix Exponential Methods

8.1 Matrix Exponentials and Linear Systems

8.2 Nonhomogeneous Linear Systems

8.3 Spectral Decomposition Methods

CHAPTER 9. Nonlinear Systems and Phenomena

9.1 Stability and the Phase Plane

9.2 Linear and Almost Linear Systems

9.3 Ecological Models: Predators and Competitors

9.4 Nonlinear Mechanical Systems

CHAPTER 10. Laplace Transform Methods

10.1 Laplace Transforms and Inverse Transforms

10.2 Transformation of Initial Value Problems

10.3 Translation and Partial Fractions

10.4 Derivatives, Integrals, and Products of Transforms

10.5 Periodic and Piecewise Continuous Input Functions

CHAPTER 11. Power Series Methods

11.1 Introduction and Review of Power Series

11.2 Power Series Solutions

11.3 Frobenius Series Solutions

11.4 Bessel Functions

References for Further Study

Appendix A: Existence and Uniqueness of Solutions

Appendix B: Theory of Determinants

Answers to Selected Problems

Index

APPLICATION MODULES

The modules listed here follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Maple, Mathematica, and MATLAB versions of these investigations are included in the Applications Manual that accompanies this textbook.

1.3 Computer-Generated Slope Fields and Solution Curves

1.4 The Logistic Equation

1.5 Indoor Temperature Oscillations

1.6 Computer Algebra Solutions

2.1 Logistic Modeling of Population Data

2.3 Rocket Propulsion

2.4 Implementing Euler's Method

2.5 Improved Euler Implementation

2.6 Runge-Kutta Implementation

3.2 Automated Row Operations

3.3 Automated Row Reduction

3.5 Automated Solution of Linear Systems

5.1 Plotting Second-Order Solution Families

5.2 Plotting Third-Order Solution Families

5.3 Approximate Solutions of Linear Equations

5.5 Automated Variation of Parameters

5.6 Forced Vibrations and Resonance

7.1 Gravitation and Kepler's Laws of Planetary Motion

7.3 Automatic Calculation of Eigenvalues and Eigenvectors

7.4 Earthquake-Induced Vibrations of Multistory Buildings

7.5 Defective Eigenvalues and Generalized Eigenvectors

7.6 Comets and Spacecraft

8.1 Automated Matrix Exponential Solutions

8.2 Automated Variation of Parameters

9.1 Phase Portraits and First-Order Equations

9.2 Phase Portraits of Almost Linear Systems

9.3 Your Own Wildlife Conservation Preserve

9.4 The Rayleigh and van der Pol Equations

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