Solving of some nonlinear ordinary differential equations in the form of power series

Abstract

In the current scientific literature, a variety of nonlinear ordinary differential equations are widely and successfully used to describe real processes in various fields of natural sciences: optics, elasticity theory, molecular physics, etc. For example, the Ermakov and Riccati equations are used to solve the quantum Schrodinger equation, in electrodynamics. However, unfortunately, there are no well-and reliably developed and generally accepted methods for solving nonlinear differential equations. In addition, most of the Riccati equations are not integrated even in quadratures. In this paper, to construct solutions to the nonlinear Ermakov and Riccati equations, it is proposed to use the corresponding so-called connected linear differential equations, the solutions of the latter are in the form of power series using modern computer systems of analytical calculations.In this paper, solutions for some nonlinear Ermakov and Riccati equations are calculated using this proposed method. It is shown by direct substitution that the obtained solutions in the form of power series satisfy the considered nonlinear equations of Ermakov and Riccati with a known accuracy. Solutions of nonlinear Ermakov and Riccati equations can be used to describe the chemical and physical properties of nanostructures at the quantum level. Besides, solutions of nonlinear Ermakov and Riccati equations can be successfully applied in solving stationary and time-dependent Schrodinger equations.

Keywords

• ordinary differential equations
• ermakov equation
• riccati equation
• mathematical modeling
• power series
• maple computer system