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Introduction to Differential Equations (Hardcover)
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Introduction to Differential Equations (Hardcover)
Series: Pure & Applied Undergraduate Texts
Expected to ship within 12 - 17 working days
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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.
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