The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis.

The first chapter treats single **differential equations,** emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton''s laws. The first order linear theory starts with a self-contained presentation of the exponential and trigonometric functions, which plays a central role in the subsequent development of this chapter. Chapter 2 provides a mini-course on linear algebra, giving detailed treatments of linear transformations, determinants and invertibility, eigenvalues and eigenvectors, and generalized eigenvectors. This treatment is more detailed than that in most differential equations texts, and provides a solid foundation for the next two chapters. Chapter 3 studies linear systems of differential equations. It starts with the matrix exponential, melding material from Chapters 1 and 2, and uses this exponential as a key tool in the linear theory. Chapter 4 deals with nonlinear systems of differential equations. This uses all the material developed in the first three chapters and moves it to a deeper level. The chapter includes theoretical studies, such as the fundamental existence and uniqueness theorem, but also has numerous examples, arising from Newtonian physics, mathematical biology, electrical circuits, and geometrical problems. These studies bring in variational methods, a fertile source of nonlinear systems of differential equations. The reader who works through this book will be well prepared for advanced studies in dynamical systems, mathematical physics, and partial differential equations.

Michael E. Taylor, *University** of North Carolina, Chapel Hill, NC*

*Preface*

**Chapter 1. Single Differential Equations**

1. The exponential and trigonometric functions

2. First order linear equations

3. Separable equations

4. Second order equations – reducible cases

5. Newton's equations for motion in 1D

6. The pendulum

7. Motion with resistance

8. Linearization

9. Second order constant coefficient linear equations – homogeneous

10. Nonhomogeneous equations I – undetermined coefficients

11. Forced pendulum – resonance

12. Spring motion

13. RLC circuits

14. Nonhomogeneous equations II – variation of parameters

15. Variable coefficient second order equations

16. Higher order linear equations

A. Where Bessel functions come from

Chapter 2. Linear Algebra

Chapter 2. Linear Algebra

1. Vector spaces

2. Linear transformations and matrices

3. Basis and dimension

4. Matrix representation of a linear transformation

5. Determinants and invertibility

6. Eigenvalues and eigenvectors

7. Generalized eigenvectors and the minimal polynomial

8. Triangular matrices

9. Inner products and norms

10. Norm, trace, and adjoint of a linear transformation

11. Self-adjoint and skew-adjoint transformations

12. Unitary and orthogonal transformations

A. The Jordan canonical form

B. Schur's upper triangular representation

C. The fundamental theorem of algebra

Chapter 3. Linear Systems of Differential Equations

Chapter 3. Linear Systems of Differential Equations

1. The matrix exponential

2. Exponentials and trigonometric functions

3. First order systems derived from higher order equations

4. Nonhomogeneous equations and Duhamel's formula

5. Simple electrical circuits

6. Second order systems

7. Curves in R3 and the Frenet-Serret equations

8. Variable coefficient systems

9. Variation of parameters and Duhamel's formula

10. Power series expansions

11. Regular singular points

A. Logarithms of matrices

**Chapter 4. Nonlinear Systems of Differential Equations**

1. Existence and uniqueness of solutions

2. Dependence of solutions on initial data and other parameters

3. Vector fields, orbits, and flows

4. Gradient vector fields

5. Newtonian equations

6. Central force problems and two-body planetary motion

7. Variational problems and the stationary action principle

8. The brachistochrone problem

9. The double pendulum

10. Momentum-quadratic Hamiltonian systems

11. Numerical study – difference schemes

12. Limit sets and periodic orbits

13. Predator-prey equations

14. Competing species equations

15. Chaos in multidimensional systems

A. The derivative in several variables

B. Convergence, compactness, and continuity

C. Critical points that are saddles

D. Periodic solutions of x + x = e?(x)

E. A dram of potential theory

F. Brouwer's fixed-point theorem

Bibliography

*Index*