A constraction of general solution for a class of non-homogeneous recurrence of variable coefficients with two indices

2000 ◽  
Vol 5 (4) ◽  
pp. 379-385
Author(s):  
Yu Chang-an
Keyword(s):  
General Solution ◽  
Variable Coefficients ◽  
Two Indices

EXPLICIT TRANSFORMATION OF THE RICCATI EQUATION AND OTHER POLYNOMIAL ODES TO SYSTEMS OF LINEAR ODES

2021 ◽  
pp. 5-14
Author(s):  
M.L. Zaytsev ◽  
V.B. Akkerman ◽  
Keyword(s):  
Linear System ◽  
General Solution ◽  
Riccati Equation ◽  
First Integral ◽  
Transformation Method ◽  
Polynomial Systems ◽  
Variable Coefficients ◽  
Particular Solution ◽  
Systems Of Odes ◽  
Ode Systems

The purpose of this work is to propose and demonstrate a way to explicitly transform polynomial ODE systems to linear ODE systems. With the help of an additional first integral, the one-dimensional Riccati equation is transformed to a linear system of three ODEs with variable coefficients. Solving the system, we can find a solution to the original Riccati equation in the general form or only to the Cauchy problem. The Riccati equation is one of the most interesting nonlinear first order differential equations. It is proved that there is no general solution of the Riccati equation in the form of quadratures; however, if at least one particular solution is known, then its general solution is also found. Thus, it is enough only to find a particular solution of the linear system of ODEs. The applied transformation method is a special case of the method described in our work [Zaytsev M. L., Akkerman V. B. (2020) On the identification of solutions to Riccati equation and the other polynomial systems of ODEs // preprint, Research Gate. DOI: 10.13140 / RG.2.2.26980.60807]. This method uses algebraic transformations and transition to new unknowns consisting of products of the original unknowns. The number of new unknowns becomes less than the number of equations. For the multidimensional Riccati equations, we do not present the corresponding linear system of ODEs because of the large number of linear equations obtained (more than 100). However, we present the first integral with which this can be done. In this paper, we also propose a method for finding the first integral, which can be used to reduce a search for the solution of any polynomial systems of ODEs to a search of solutions to linear systems of ODEs. In particular, if the coefficients in these equations are constant, then the solution is found explicitly.

scholarly journals Elastic transformation method for solving ordinary differential equations with variable coefficients

2021 ◽  
Vol 7 (1) ◽  
pp. 1307-1320
Author(s):  
Pengshe Zheng ◽  
◽  
Jing Luo ◽  
Shunchu Li ◽  
Xiaoxu Dong
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transformation Method ◽  
Variable Coefficients ◽  
Nonlinear Ordinary Differential Equations ◽  
Third Order ◽  
First Order ◽  
First Time

<abstract><p>Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.</p></abstract>

The Role of VLBI in the Establishment of Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 293-295 ◽  
Author(s):  
I. Zhongolovitch
Keyword(s):  
General Solution ◽  
Future Development ◽  
Rational Approach ◽  
Radio Telescopes ◽  
Coordinate Systems ◽  
Tectonic Plates ◽  
Astronomical Observations ◽  
Optical Instruments ◽  
Fundamental Polyhedron

Considering the future development and general solution of the problem under consideration and also the high precision attainable by astronomical observations, the following procedure may be the most rational approach:1. On the main tectonic plates of the Earth’s crust, powerful movable radio telescopes should be mounted at the same points where standard optical instruments are installed. There should be two stations separated by a distance of about 6 to 8000 kilometers on each plate. Thus, we obtain a fundamental polyhedron embracing the whole Earth with about 10 to 12 apexes, and with its sides represented by VLBI.

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