Fontaine’s Forgotten Method for Inexact Differential Equations

Abstract

SummaryIn 1739 Alexis-Claude Clairaut published the modern integrating factor method of solving inexact ordinary differential equations (ODEs). He was motivated by a 1738 Alexis Fontaine paper with a different method which requires solving a difficult partial differential equation (PDE). Here we revisit Fontaine's method, examine his modest attempt to solve the PDE, and utilize a different technique to give (we believe) the first family of ODEs solvable by Fontaine's method with no obvious solution using the modern technique.