George Simmonss fine text is a traditional book written in the classical style. It provides a cogent and accessible introduction to all the traditional topics. It is a pleasure to have this opportunity to bring this text up to date and to add some more timely material. We have streamlined some of the exposition and augmented other parts. There is now computer work based not only on number crunching but also on computer algebra systems such as Maple， Mathematica， and MATLAB. Certainly a study of flows and vector fields， and of the beautiful Poincare-Bendixson theory built theorem， is essential for any modern treatment. One can introduce some of the modern ideas from the theory of dynamics to obtain qualitative information about nonlinear differential equations and systems.And all of the above is a basis for modeling. Modeling is what brings the subject to life and makes the ideas real for the students： Differential equations can model real-life questions，and computer calculations and graphics can then provide real-life answers. The symbiosis of the synthetic and the calculational provides a rich educational experience for students， and it prepares them for more concrete， applied work in future courses. The new Anatomy of an Application sections in this edition showcase some rich applications from engineering， physics， and applied science.
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