PART 1 Ordinary Differential Equations

1 Introduction to Differential Equations

1.1 Definitions and Terminology

1.2 Initial-Value Problems

1.3 Differential Equations as Mathematical Models

Chapter 1 in Review

2 First-Order Differential Equations

2.1 Solution Curves Without a Solution

2.1.1 Direction Fields

2.1.2 Autonomous First-Order DEs

2.2 Separable Equations

2.3 Linear Equations

2.4 Exact Equations

2.5 Solutions by Substitutions

2.6 A Numerical Method

2.7 Linear Models

2.8 Nonlinear Models

2.9 Modeling with Systems of First-Order DEs

Chapter 2 in Review

3 Higher-Order Differential Equations

3.1 Theory of Linear Equations

3.1.1 Initial-Value and Boundary-Value Problems

3.1.2 Homogeneous Equations

3.1.3 Nonhomogeneous Equations

3.2 Reduction of Order

3.3 Homogeneous Linear Equations with Constant Coefficients

3.4 Undetermined Coefficients

3.5 Variation of Parameters

3.6 Cauchy–Euler Equations

3.7 Nonlinear Equations

3.8 Linear Models: Initial-Value Problems

3.8.1 Spring/Mass Systems: Free Undamped Motion

3.8.2 Spring/Mass Systems: Free Damped Motion

3.8.3 Spring/Mass Systems: Driven Motion

3.8.4 Series Circuit Analogue

3.9 Linear Models: Boundary-Value Problems

3.10 Green’s Functions

3.10.1 Initial-Value Problems

3.10.2 Boundary-Value Problems

3.11 Nonlinear Models

3.12 Solving Systems of Linear Equations

Chapter 3 in Review

4 The Laplace Transform

4.1 Definition of the Laplace Transform

4.2 The Inverse Transform and Transforms of Derivatives

4.2.1 Inverse Transforms

4.2.2 Transforms of Derivatives

4.3 Translation Theorems

4.3.1 Translation on the s-axis

4.3.2 Translation on the t-axis

4.4 Additional Operational Properties

4.4.1 Derivatives of Transforms

4.4.2 Transforms of Integrals

4.4.3 Transform of a Periodic Function

4.5 The Dirac Delta Function

4.6 Systems of Linear Differential Equations

Chapter 4 in Review

5 Series Solutions of Linear Differential Equations

5.1 Solutions about Ordinary Points

5.1.1 Review of Power Series

5.1.2 Power Series Solutions

5.2 Solutions about Singular Points

5.3 Special Functions

5.3.1 Bessel Functions

5.3.2 Legendre Functions

Chapter 5 in Review

6 Numerical Solutions of Ordinary Differential Equations

6.1 Euler Methods and Error Analysis

6.2 Runge–Kutta Methods

6.3 Multistep Methods

6.4 Higher-Order Equations and Systems

6.5 Second-Order Boundary-Value Problems

Chapter 6 in Review

PART 2 Vectors, Matrices, and Vector Calculus

7 Vectors

7.1 Vectors in 2-Space

7.2 Vectors in 3-Space

7.3 Dot Product

7.4 Cross Product

7.5 Lines and Planes in 3-Space

7.6 Vector Spaces

7.7 Gram–Schmidt Orthogonalization Process

Chapter 7 in Review

8 Matrices

8.1 Matrix Algebra

8.2 Systems of Linear Algebraic Equations

8.3 Rank of a Matrix

8.4 Determinants

8.5 Properties of Determinants

8.6 Inverse of a Matrix

8.6.1 Finding the Inverse

8.6.2 Using the Inverse to Solve Systems

8.7 Cramer’s Rule

8.8 The Eigenvalue Problem

8.9 Powers of Matrices

8.10 Orthogonal Matrices

8.11 Approximation of Eigenvalues

8.12 Diagonalization

8.13 LU-Factorization

8.14 Cryptography

8.15 An Error-Correcting Code

8.16 Method of Least Squares

8.17 Discrete Compartmental Models

Chapter 8 in Review

9 Vector Calculus

9.1 Vector Functions

9.2 Motion on a Curve

9.3 Curvature and Components of Acceleration

9.4 Partial Derivatives

9.5 Directional Derivative

9.6 Tangent Planes and Normal Lines

9.7 Curl and Divergence

9.8 Line Integrals

9.9 Independence of the Path

9.10 Double Integrals

9.11 Double Integrals in Polar Coordinates

9.12 Green’s Theorem

9.13 Surface Integrals

9.14 Stokes’ Theorem

9.15 Triple Integrals

9.16 Divergence Theorem

9.17 Change of Variables in Multiple Integrals

Chapter 9 in Review

PART 3 Systems of Differential Equations

10 Systems of Linear Differential Equations

10.1 Theory of Linear Systems

10.2 Homogeneous Linear Systems

10.2.1 Distinct Real Eigenvalues

10.2.2 Repeated Eigenvalues

10.2.3 Complex Eigenvalues

10.3 Solution by Diagonalization

10.4 Nonhomogeneous Linear Systems

10.4.1 Undetermined Coefficients

10.4.2 Variation of Parameters

10.4.3 Diagonalization

10.5 Matrix Exponential

Chapter 10 in Review

11 Systems of Nonlinear Differential Equations

11.1 Autonomous Systems

11.2 Stability of Linear Systems

11.3 Linearization and Local Stability

11.4 Autonomous Systems as Mathematical Models

11.5 Periodic Solutions, Limit Cycles, and Global Stability

Chapter 11 in Review

PART 4 Partial Differential Equations

12 Orthogonal Functions and Fourier Series

12.1 Orthogonal Functions

12.2 Fourier Series

12.3 Fourier Cosine and Sine Series

12.4 Complex Fourier Series

12.5 Sturm–Liouville Problem

12.6 Bessel and Legendre Series

12.6.1 Fourier–Bessel Series

12.6.2 Fourier–Legendre Series

Chapter 12 in Review

13 Boundary-Value Problems in Rectangular Coordinates

13.1 Separable Partial Differential Equations

13.2 Classical PDEs and Boundary-Value Problems

13.3 Heat Equation

13.4 Wave Equation

13.5 Laplace’s Equation

13.6 Nonhomogeneous Boundary-Value Problems

13.7 Orthogonal Series Expansions

13.8 Fourier Series in Two Variables

Chapter 13 in Review

14 Boundary-Value Problems in Other Coordinate Systems

14.1 Polar Coordinates

14.2 Cylindrical Coordinates

14.3 Spherical Coordinates

Chapter 14 in Review

15 Integral Transform Method

15.1 Error Function

15.2 Applications of the Laplace Transform

15.3 Fourier Integral

15.4 Fourier Transforms

15.5 Fast Fourier Transform

Chapter 15 in Review

16 Numerical Solutions of Partial Differential Equations

16.1 Laplace’s Equation

16.2 Heat Equation

16.3 Wave Equation

Chapter 16 in Review

PART 5 Complex Analysis

17 Functions of a Complex Variable

17.1 Complex Numbers

17.2 Powers and Roots

17.3 Sets in the Complex Plane

17.4 Functions of a Complex Variable

17.5 Cauchy–Riemann Equations

17.6 Exponential and Logarithmic Functions

17.7 Trigonometric and Hyperbolic Functions

17.8 Inverse Trigonometric and Hyperbolic Functions

Chapter 17 in Review

18 Integration in the Complex Plane

18.1 Contour Integrals

18.2 Cauchy–Goursat Theorem

18.3 Independence of the Path

18.4 Cauchy’s Integral Formulas

Chapter 18 in Review

19 Series and Residues

19.1 Sequences and Series

19.2 Taylor Series

19.3 Laurent Series

19.4 Zeros and Poles

19.5 Residues and Residue Theorem

19.6 Evaluation of Real Integrals

Chapter 19 in Review

20 Conformal Mappings

20.1 Complex Functions as Mappings

20.2 Conformal Mappings

20.3 Linear Fractional Transformations

20.4 Schwarz–Christoffel Transformations

20.5 Poisson Integral Formulas

20.6 Applications

Chapter 20 in Review

Appendices

Appendix I: Derivative and Integral Formulas

Appendix II: Gamma Function

Appendix III: Table of Laplace Transforms

Appendix IV: Conformal Mappings

Answers to Selected Odd-Numbered Problems

Index