A formula of general solution for a class of homogeneous trinomial recurrence of variable coefficients with two indices

2005 ◽  
Vol 10 (5) ◽  
pp. 828-832
Author(s):  
Yu Dan ◽  
Yu Chang-an
Keyword(s):  
General Solution ◽  
Variable Coefficients ◽  
Two Indices ◽  
Formula Of General Solution

scholarly journals The general solution of impulsive systems with Riemann-Liouville fractional derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1125-1137 ◽  
Author(s):  
Xianmin Zhang ◽  
Wenbin Ding ◽  
Hui Peng ◽  
Zuohua Liu ◽  
Tong Shu
Keyword(s):  
General Solution ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Fractional Order System ◽  
Order System ◽  
Impulsive Effect ◽  
Impulsive Systems ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Formula Of General Solution

AbstractIn this paper, we study a kind of fractional differential system with impulsive effect and find the formula of general solution for the impulsive fractional-order system by analysis of the limit case (as impulse tends to zero). The obtained result shows that the deviation caused by impulses for fractional-order system is undetermined. An example is also provided to illustrate the result.

scholarly journals On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)

2016 ◽  
Vol 14 (1) ◽  
pp. 452-473 ◽  
Author(s):  
Xianmin Zhang ◽  
Tong Shu ◽  
Zuohua Liu ◽  
Wenbin Ding ◽  
Hui Peng ◽  
...  
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Impulsive Differential Equations ◽  
Formula Of General Solution

AbstractIn this paper, we find the formula of general solution for a generalized impulsive differential equations of fractional-order q ∈ (2, 3).

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.

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