Introduction

In many ways, differential equations are the heart of analysis and calculus, two of the most important branches of mathematics for over the past 300 years. Differential equations are an integral part or a goal of many undergraduate calculus courses. As an important mathematical tool for the physical sciences, the differential equation has no equal. So it is widely accepted that differential equations are important in both applied and pure mathematics.

The foundations of this subject seem to be so dominated by the contributions of one man, Leonhard Euler, that one could say the history of this subject starts and ends with him.

Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.

Of course, that would be a gross simplification of its development. There are many important contributors, and those who came before Euler were necessary so that Euler could understand the calculus and analysis necessary to develop many of the fundamental ideas. The contributors after Euler have both refined his work and forged entirely new ideas, inaccessible to Euler's 18th-century perspective and sophisticated beyond the understanding of just one person.