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Singularities of Singular Solutions of First-Order Differential Equations of Clairaut Type Kentaro Saji1 · Masatomo Takahashi2 Received: 28 April 2020 / Revised: 3 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract A first-order differential equation of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. The projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). In these cases, the envelopes are always fronts. We investigate singular points of envelopes for first-order ordinary differential equations, first-order partial differential equations, and systems of first-order partial differential equations of Clairaut type, respectively. Keywords Clairaut type · Singular solution · Envelope · Singular point PACS 58K05 · 57R45 · 34A09 · 35A09

1 Introduction Recently, the curvature of Legendre curves (cf. [6]) and criteria of generic singularities are known (cf. [5, 16, 17, 19, 21]). In addition, studies of envelopes were also conducted (cf. [24, 25]). In this paper, we discuss first-order ordinary differential equations, first-order partial differential equations, and systems of first-order partial differential equations of Clairaut type as these applications. A first-order differential equation of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. Then, the projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). We investigate properties of envelopes. In these cases, the envelopes are always fronts (Propositions 3.2, 4.2, and 5.2). We investigate singular points of envelopes of Clairaut type. The first author was a partially supported by JSPS KAKENHI Grant Number JP 18K03301. The second author was a partially supported by JSPS KAKENHI Grant Numbers JP 17K05238 and 20K03573.  Masatomo Takahashi

[email protected] Kentaro Saji [email protected] 1

Department of Mathematics, Kobe University, Kobe, 657-8501, Japan

2

Muroran Institute of Technology, Muroran, 050-8585, Japan

Kentaro Saji and Masatomo Takahashi

In Section 2, we review on the theories of Legendre curves, one-parameter families of Legendre curves, and Legendre surfaces. In Section 3, we consider first-order ordinary differential equations of the Clairaut type. We give conditions that envelopes of the projection of classical complete solutions have 3/2 and 4/3 cusps. In Section 4, we consider first-order partial differential equations of the Clairaut type of two variables. We give conditions that envelopes of the projection of classical complete solutions have cuspidal edge, swallowtail, cuspidal butterfly, cuspidal lips, cuspidal beaks, and D4± singular points. In Section 5, we consider systems of first-order partial differential equations of the Clairaut type of two variables. In this case, the corank of the envelope of the projection of a classical comple