Introduction

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Will this airplane fly?... How can we create an image of the interior of the human body using very weak X-rays?... What is a design of a bicycle frame that combines low weight with rigidity?... How much would the mean temperature of the earth increase if the amount of carbon dioxide in the atmosphere increased by 20 percent?

—An overview of applications of differential equations in Computational Differential Equations, K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Cambridge University Press, New York, 1996

Tour de France champion Lance Armstrong testing a bicycle at the San Diego Air & Space Technology Low Speed Wind Tunnel in November 2008. Armstrong's clothing, helmet, posture, and hand position are also aerodynamically optimized.

Differential equations are among the most powerful tools we have for analyzing the world mathematically. They are used to formulate the fundamental laws of nature (from Newton's Laws to Maxwell's equations and the laws of quantum mechanics) and to model the most diverse physical phenomena. The quotation above lists just a few of the myriad applications. This chapter provides an introduction to some elementary techniques and applications of this important subject.