On the history of the theory of linear differential equations

Abstract

I shall discuss the remarkable analogy between algebraic and linear differential equations, an analogy which largely determined the development of the theory of the latter throughout the 19th century. This was a heroic period in the history of the theory of algebraic equations. The centuries-old development of the theory, which mainly determined the range of algebra as a whole, was in a sense completed by a series of outstanding achievements (by Abel, Galois, and others). In any case, even in the second half of the 19th century, J. A. Serret [1, p. 1] in introducing the subject of his well known Cours d'algebre superieure, stated that L'algebre est, proprement parler, l'Analyse des equations. By the beginning of our century such opinions had become obsolete. A new understanding of algebra as a science of algebraic structures (N. Bourbaki) was taking shape and the theory of algebraic equations itself was little by little downgraded to a chapter in university algebra courses. In the 19th century, however, algebraic equations commanded general attention; mathematicians of the