27 - The Origins of Integrating Factors

Summary

Introduction

In a differential equations course, students learn to use integrating factors to solve first order linear differential equations, and in the process reinforce learning of key concepts from their calculus courses. This capsule offers some differential equations solved by the originators of the technique of using an integrating factor, though they did not use that expression. Solving differential equations via integrating factors is difficult for some students, particularly those who try to memorize a formula. We advocate that students learn to derive the method and solve differential equations using the product rule and the fundamental theorem of calculus, as advocated in a number of modern texts [2, 3]. Memorizing the formula would not be in the spirit of the originators of the method, Johann Bernoulli (1667–1748) and Leonhard Euler (1707–1783), nor does formula memorization lead to deep learning of fundamental mathematical processes. Understanding why integrating factors work, as offered in this historical perspective, can deepen student understanding of calculus topics such as the product rule, the fundamental theorem of calculus, and basic integration techniques. This capsule provides some historical information about the work of Bernoulli and Euler, and we offer student activities that will connect that history to enable more thorough learning of the method of integrating factors.

Historical preliminaries

Johann Bernoulli was a colleague of Gottfried Leibniz (1646–1716) and is acknowledged as one of the foundational figures in the development of the calculus. In the early 1690's he prepared lectures in the nascent calculus for Guillaume de l'Hôpital (1661–1704), who is credited with writing the first text on the calculus.