Dennis G. Zill, Loyola Marymount University, Michael R. Cullen, Late, Loyola Marymount University

Part I: Ordinary Differential Equations

Chapter 1. Introduction to Differential Equations
1.1. Definitions and Terminology
1.2. Initial-Value Problems
1.3. Differential Equations as Mathematical Models

Chapter 1. Review Exercises
Chapter 2. First-Order Differential Equations
2.1. Solution Curves Without the Solution
2.2. Separable Variables
2.3. Linear Equations
2.4. Exact Equations
2.5. Solutions by Substitutions
2.6. A Numerical Solutions
2.7. Linear Models
2.8. Nonlinear
2.9. Systems: Linear and Nonlinear Models
Chapter 2. Review Exercises

Chapter 3. Higher-Order Differential Equations
3.1. Preliminary Theory: Linear and Nonlinear Models
3.2. Reduction of Order
3.3. Homogenous Linear Equations with Constant Coefficients
3.4. Undetermined
3.5. Variations of Parameters
3.6. Cauchy-Euler Equation
3.7. Nonlinear Equations
3.8. Linear Models: Initial-Value Problems
3.9. Linear Models: Boundary-Value Problems
3.10. Nonlinear Models
3.11. Solving Systems of Linear Models
Chapter 3. Review Exercises

Chapter 4. The Laplace Transform
4.1. Definition of the Laplace Transform
4.2. The Inverse Transform and Transforms of Derivations
4.3. Translation Theorems
4.5. Dirac Delta Function
4.6. Solving Systems of Linear Equations
Chapter 4. Review Exercises

Chapter 5. Series Solutions of Linear Equations
5.3. Two Special Equations
Chapter 5. Review Exercises

Chapter 6. Numerical Solutions of Ordinary Differential Equations
6.1. Euler Methods and Error Analysis
6.2. Runge-Kutta Methods
6.3. Methods
6.4. Higher-Order Equations and Systems
6.5. Second-Order Boundary-Value Problems

Part II: Vectors, Matrices, and Vector Calculus

Chapter 7. Vectors
7.1. Vectors in 2-Space
7.2. Vectors in 3-Space
7.3. The Dot Product
7.4. The Cross Product
7.5. Lines and Planes in 3-Space
7.6. Vector Spaces
Chapter 7. Review Exercises

Chapter 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
8.7. Cramer's Rule
8.8. The Eigenvalue Problem
8.9. Power of Matrices
8.10. Orthogonal Matrices
8.11. Approximation of Eigenvalues
8.12. Diagonalization
8.13. Cryptography
8.14. An Error-Correcting Code
8.15. Method of Least Squares
8.16. Discrete Compartmental Models
Chapter 8. Review Exercises

Chapter 9. Vector Calculus
9.1. Vector Functions
9.2. Motion on a Curve
9.3. Curvature and Components of Acceleration
9.4. Functions of Several Variables
9.5. The Directional Derivative
9.6. Planes and Normal Lines
9.7. Divergence and Curl
9.8. Line Integrals
9.9. Line Integrals Independent of the Path
9.10. Review of Double Integrals
9.11. Double Integrals in Polar Coordinates
9.12. Green's Theorem
9.13. Surface Integrals
9.14. Strokes' Theorem
9.15. Review of Triple Integrals
9.16. Divergence Theorem
9.17. Change of Variables in Multiple Integrals
Chapter 9. Review Exercises

Part III: Systems of Differential Equations

Chapter 10. System of Linear Differential Equations
10.1. Preliminary
10.2. Homogeneous Linear Systems
10.3. Solution by Diagonalization
10.4. Nonhomogenous Linear Systems
10.5. Matrix Exponential
Chapter 10. Review Exercise

Chapter 11. Systems of Nonlinear Differential Equations
11.1. Autonomous Systems, Critical Points, and Periodic Solutions
11.2. Stability of Linear Systems
11.3. Linearization and Local Stability
11.4. Modeling Using Autonomous Systems
11.5. Periodic Solutions, Limit Cycles, and Global Stability
Chapter 11. Review Exercise

Part IV: Fourier Series and Partial Differential Equations

Chapter 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 and Frequency Spectrum
12.5. Sturm-Liouville Problem
12.6. Bessel and Legendre Series
Chapter 12. Review Exercises

Chapter 13. Boundary-Value Problems in Rectangular Coordinates
13.1. Separable Partial Differential Equations
13.2. Classical Equations and Boundary-Value Problems
13.3. Heat Equation
13.4. Wave Equation
13.5. Laplace's Equation
13.6. Nonhomogeneous Equations and Boundary Conditions
13.7. Orthogonal Series Expansions
13.8. Fourier Series in Two Variable
Chapter 13. Review Exercises

Chapter 14. Boundary-Value Problems in Other Coordinate Systems
14.1. Problems Involving Laplace's Equation in Polar Coordinates
14.2. Problems in Polar and Cylindrical Coordinates: Bessel Functions
14.3. Problems in Spherical Coordinates: Legendre Polynomials
Chapter 14. Review Exercises

Chapter 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. Review Exercises

Chapter 16. Numerical Solutions to Partial Differential Equations
16.1. Elliptic Equations
16.2. Parabolic Equations
16.3. Hyperbolic Equations
Chapter 16. Review Exercises

Part V: Complex Analysis

Chapter 17. Functions of a Complex Variable
17.1. Complex Numbers
17.2. Form of Complex Numbers; Power and Roots
17.3. Sets of Points in the Complex Plane
17.4. Functions of a Complex Variable; Analyticity
17.5. Cauchy-Reimann Equations
17.6. Exponential and Logarithmic Functions
17.7. Trigonometric and Hyperbolic Functions
17.8. Inverse Trigonometric and Hyperbolic Functions
Chapter 17. Review Exercise

Chapter 18. Integration in the Complex Plane
18.1. Contour Integrals
18.2. Cauchy-Goursat Theorem
18.3. Independence of Path
18.4. Cauchy's Integral Formula
Chapter 18. Review Exercises

Chapter 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. Review Exercises

Chapter 20. Conformal Mappings and Applications
20.1. Complex Functions as Mappings
20.2. Conformal Mapping and the Dirichlet Problem
20.3. Linear Fractional Transformations
20.4. Schwarz-Christoffel Transformations
20.5. Poisson Integral Formulas
20.6. Applications
Chapter 20. Review Exercise

Appendix I Some Derivative and Integral Formulas
Appendix II Gamma Function; Exercises
Appendix III Table of Laplace Transforms
Appendix IV Conformal Mappings
Appendix V Some BASIC Programs for Numerical Methods