The properties of certain linear and nonlinear differential equations of the fourth order arising in beam models

Abstract

The purpose of this paper is to present several new results concerning relations between linear differential equations of the fourth order and nonlinear differential equations of the fourth order. These equations are involved in the description of models of building structures, where there are beams with small deflections or curved axes. We consider linear differential equations of the second, the third and the fourth order and nonlinear fourth order differential equations related via the Schwarzian derivative. As a result, we obtain new relations between the solutions of these linear and nonlinear equations. For example, assuming that we know a solution of linear differential equation of the fourth order and a solution of the third order linear differential equation, then the Schwarzian derivative of their ratio solves certain nonlinear differential equation of the fourth order. We also present some conditions on the coefficients when this statement holds. Two more similar statements are presented. To illustrate theorems and our constructive approach we give two examples. The given method may be generalized to differential equations of higher orders.