ABSTRACT
A differential equation is an equation which involves an unknown function and its derivatives. This chapter considers ordinary differential equations which concern only functions of one independent variable. Differential equations have been studied since Newton invented the calculus, which means that people have worked on them for more than 300 years. The chapter describes some techniques which were developed mainly in the 1700s and 1800s but which are still often useful. A geometrical approach yields a good qualitative understanding not only of particular solutions but also of the relationships among the solutions. By the end of the 1800s, mathematicians had, largely by struggling with nonlinear equations in celestial mechanics, become aware of the deep problems inherent in the study of nonlinear equations. The result was that various techniques were introduced which are used to investigate different aspects or properties of solutions.
