Differential Equations: A Concise Course (Dover Books on Mathematics)

This concise treatment of differential equations is intended to serve as a text for a standard one-semester or two-term undergraduate course in differential equations following the calculus. Emphasis is placed on mathematical explanations — ranging from routine calculations to moderately sophisticated theorems — in order to impart more than a rote understanding of techniques.
Beginning with a survey of first order equations, the text goes on to consider linear equations — including discussions of complex-valued solutions, linear differential operators, inverse operators, and variation of parameters method. Subsequent chapters then examine the Laplace transform and Picard's existence theorem and conclude with an exploration of various interpretations of systems of equations.
Numerous clearly stated theorems and proofs, examples, and problems followed by solutions make this a first-rate introduction to differential equations.