A Singular Integral for a Simplified Clairaut Equation

TL;DR

The paper investigates the simplified Clairaut equation $x z_x + y z_y = z$ by connecting Euler's homogeneous function solutions to Lagrange's complete integral $z=ax+by$ and extending Goursat's general integral. It shows that Euler's degree-1 homogeneous solutions arise as envelopes of planar families obtained from the complete integral, and further demonstrates that noninvertible and generalized relations between $a$ and $b$ yield broader, nonfunctional surfaces that go beyond Euler. Three constructions are developed to bridge the general integral and the complete integral: (i) a standard general integral with $b= ext{phi}(a)$, (ii) a generalized general integral with $ ext{phi}(a,b)=0$, and (iii) a parametric or inverse-map approach leading to tilted cone surfaces; together they partially address Evans and Chojnacki's critiques about the scope and behavior of general and complete integrals. The work emphasizes careful envelope calculations, discusses cusp-related limitations, and frames the results in a differential-geometric language that connects tangent plane distributions with integral surfaces, suggesting directions for future work in Frobenius-type integrability. $x z_x + y z_y = z$ thus serves as a pedagogical testbed for core PDE concepts including complete/general/singular integrals and their geometric interpretations.

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.

Figures (8)

• Figure 1: Integrals for $y'^2 = 4y$. The family of parabolas $y = (x + c) ^2$ is called the complete integral, and each member of this family satisfies the PDE. A curve tangent to this family of parabolas is the $x$ axis $y = 0$, and it is called the singular integral, and it satisfies the PDE as well.
• Figure 2: $ax + by = 1$ (with $ab = 1$), $4xy = 1$
• Figure 3: Starting with $y^4 - y^2 - (x - a)^2 = 0$, the line $y=0$ is a locus of singularities, while $y=\pm 1$ represents an envelope
• Figure 4: Cone as the envelope of planes
• Figure 5: The graphs on the left represent various functions of $a$ starting with an explicitly constructed $\phi'(a)$ that is non-invertible by design. Given $\phi(a)$ as defined on the left, the graph on the right shows a cross section of the surface $z = ax + \phi(a)y$ at $z=1$. $P$ and $Q$ lie on the same radial line, and hence have the same value of $\frac{x}{y}$ in this cross-sectional view. When we project up along the z-axis to get a cone-like surface, this translates to one to many relationship between $(x, y)$ and $z$.