Solution of the nonlinear ordinary differential equation of Ermakov by power series
DOI:
https://doi.org/10.52575/2687-0959-2022-54-3-171-177Keywords:
mathematical modeling, symbolic-numerical methods, software packages, differential equation, nonlinear ordinary differential equation of second order, equation of Ermakov, power seriesAbstract
Nonlinear differential equations are widely used in various modern sciences. In particular, the nonlinear ordinary differential equation of Ermakov is successfully used to solve problems in quantum mechanics, electrodynamics, optics, elasticity theory, to describe molecular structures, in heterostructures with a complex potential function and in many other branches of theoretical and mathematical physics. However, there is currently no effective method for solving nonlinear equations such as the Ermakov equation. For example, when solving eigenvalue problems, known modern authors calculated solutions of the Ermakov equation by direct numerical methods. As is known from the works of Ermakov himself and other modern authors, the solution of the Ermakov equation is determined by two linearly independent solutions of a suitable so-called attached linear differential equation of the second order. The theory of integration of linear differential equations by power series is mathematically strictly developed, in particular, for the attached linear equations to the Ermakov equation, the convergence of power series representing the solution of the attached linear differential equations is proved. In this paper, these linearly independent solutions of the attached linear equation were calculated in the form of power series using the MAPLE analytical computing computer system and for a number of Ermakov equations, their solutions were constructed in the form of power series, in general, with an arbitrary number terms. By direct substitution, it was shown that the power series obtained in this way satisfy the Ermakov equation. The obtained solutions in the form of power series containing also a spectral parameter can be successfully applied to solving eigenvalue problems, in particular for solving the stationary Schrodinger equation.
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