Linear equations with one constraint and their connection to nonlinear equations of the fourth order

  • Alexander Chichurin, Institute of Mathematics and Computer Science, The John Paul II Catholic University of Lublin, ul. Konstantynow 1H, 20-708 Lublin, Poland,
  • Galina Filipuk, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland,


The purpose of this paper is to present several new results concerning relations between linear differential equations of the fourth order with one constraint and nonlinear differential equations of the fourth order. 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. The method is based on the use of the Schwarzian derivative, which is defined as the ratio of two linearly independent solutions of the linear differential equations of the second or third and fourth order. As a result we obtain new relations between the solutions of these linear and nonlinear equations. To illustrate theorems and our constructive approach we give two examples. The given method may be generalized to differential equations of higher orders.

Vol. 22 (2021), No. 1, pp. 133-141
DOI: 10.18514/MMN.2021.2956

Download: MMN-2956