I would like to revisit the question of attribution for the method of
variation of parameters, which was the subject of a few email messages
several months ago.
First, thank you again for the reference to your paper and to those of
Lagrange and Cauchy. I recently had the opportunity to spend a few minutes
with the collected works of both of the latter mathematicians. In
Lagrange's Oeuvres, vol 4, pp 151-251, there is a paper entitled (in part)
"Recherches sur les suites recurrentes ..." and dated 1775. On pages 159
through 161 of this paper is what I consider a clear and detailed
description of the method of variation of parameters for a linear equation
of order n with variable coefficients.
Lagrange's presentation does not differ significantly from what appears in
modern books, such as mine (pages 221-222). The only thing lacking is an
explicit expression for the derivatives of the coefficients. Lagrange
merely says to solve the system of n linear algebraic equations for the
derivatives of the coefficients and then to integrate to find the
coefficients themselves. But apart from this the method is certainly there.
This paper, as well as a slightly earlier one (Oeuvres,4,pp. 5-108), are
both cited by Morris Kline in his "Mathematical Thought from Ancient to
Modern Times".
Kline also cites papers by Johann Bernoulli, Euler, and Laplace, but I have
had no opportunity to follow these up.
So my conclusion is that it is appropriate to attribute the method to
Lagrange rather than to Cauchy. I don't know how much credit should also
go to Bernoulli, Euler, or Laplace.
With best wishes,
Bill Boyce
W. E. Boyce
Professor Emeritus
Rensselaer Polytechnic Institute