Research into a class of third-order nonlinear differential equations in the domain of analyticity

Abstract

We consider the class of ordinary third-order nonlinear differential equations with a polynomial right-hand side of the second degree, which has movable singular points of algebraic type and is in general unsolvable in quadratures. The existing classical theory, in particular the Cauchy theorem of the existence of a solution to a differential equation, in this case is practically inapplicable. To solve this category of equations, one of the authors has developed an analytical approximate method consisting of six mathematical problems. The paper presents a study of the analytical approximate solution in the domain of analyticity, including the solution existence theorem proof, the making of an analytical approximate solution, and the investigation of the effect of initial data perturbation on the analytical approximate solution. The existence theorem proof is based on the majorant method in a new version, which makes it possible to carry out the planned investigations. A computational experiment with the use of a posteriori error estimation is presented, which makes it possible to significantly improve the a priori error estimation obtained.