Ordinary Differential Equations and Stability Theory: An Introduction (Dover Books on Mathematics)

This is a brief, modern introduction to the subject of ordinary differential equations, with an emphasis on stability theory. Concisely and lucidly expressed, it is intended as a supplementary text for the advanced undergraduate or beginning graduate student who has had a first course in ordinary differential equations.
The author begins with a general discussion of the linear equation and develops the notions of a fundamental system of solutions, the Wronskian, and the corresponding fundamental matrix. He then introduces the nonhomogeneous linear equation and the important variation of parameters formula, by which, following a consideration of the nth-order linear equation, the solution of the nonhomogeneous nth-order linear equation is obtained. A chapter is then devoted to the linear equation with constant coefficients.
The two following chapters are introductory discussions of stability theory for autonomous and nonautonomous systems. Included here are two results for nonlinear systems, Liapunov’s direct method, and some results for the second-order linear equation. The final chapter takes up the problems of the existence and uniqueness of solutions and related topics. Two appendixes — “Series Solutions of Second-Order Linear Equations” and “Linear Systems with Periodic Coefficients” — are also provided, and there are problems at the end of each chapter.