A Singular Integral for a Simplified Clairaut Equation

TL;DR

This paper explores a new connection between the Lagrange singular integral and Euler's theorem for simplified Clairaut equations, introducing a generalized integral concept that bridges general and complete integrals in PDEs.

Contribution

It introduces a generalized notion of the general integral, linking it more closely to the complete integral and addressing limitations in classical PDE solution methods.

Findings

Derived Euler's solution from Lagrange's integral with caveats

Constructed more general surfaces beyond Euler's solutions

Bridged gap between general and complete integrals

Abstract

This expository article on the Lagrange singular integral contains two novelties. The first novelty involves a connection between the Lagrange singular integral for a simplified Clairaut equation, and Euler's homogeneous function theorem. The paper presents a formal derivation of Euler's solution from Lagrange's complete integral, though with some caveats, and also constructs more general surfaces from the complete integral which go beyond Euler's solutions. The first rather complicated construction is based directly on Goursat's definition of a general integral, while the subsequent simpler constructions are based on a suitably expanded notion of the general integral. This generalized general integral is our second novelty. It bridges some of the gap between the the general integral, and the complete integral, partially addressing Evans' remarks (Partial Differential Equations, AMS Graduate Studies in Mathematics, 1998) on the limitations of the general integral. Finally we discuss some subtleties around complete integrals as noted by Chojnacki (Proceedings of the AMS, 1995) and some around general integrals as noted by Evans, and how they apply to our examples. We aim to present these classical PDE concepts to readers with a basic knowledge of multivariable calculus.