Uogólnione Równania N-go Rzędu Riccatiego Drugiego Rodzaju. Zastosowania w Teorii Sprężystości

Abstract

In some problems of the theory of elasticity (cf. [6], [12]) there appear differential equations leading to the notion of generalized n-th order Riccati equations. These are non-linear equations the theory of which is in close connection with the theory of (n+1) -st order linear equations, Ln+1. In a previous publication by the present author [9], a theory of these equations has been given assuming multiple differentiability of the coefficients of the equation. In this paper a generalized theory of Riccati equations of the second kind is given (denoted by the symbol Rn), requiring weaker
assumptions (only those of continuity of coefficients). In Sec. 1 Rn - equations are defined and several theorems are proved, concerning the properties of these equations and also their connection with the Ln+1 equations corresponding to Rn. Sec. 2 explains the role of Rn - equations in the problem of resolution of the linear differential equation Ln+1[y] into a product of operator factors. Sec. 3 contains a discussion of some applications of the theory of R, equations to the theory of linear equations. Sec. 4 brings a definition of the so called elementarly resolvable equations (the E-class). The E-class, to which belong, as the simplest particular cases, the equations with constant coefficients and the Euler equations, constitutes a natural generalization of the letter class. A general method is given for reducing the order of an E-class equation. Theoretical results are illustrated by examples.
Section 5 contains an accurate solution of the problem of an elastic beam with variable cross-section described by the function (5.2) and acted on by an arbitrary vertical load depending on x and a compressive axial force P.