Sensitivity of Hurwitz stability of linear differential equation systems with constant coefficients

2017 ◽  
Vol 14 (06) ◽  
pp. 1750084
Author(s):  
Ahmet Duman ◽  
Kemal Aydin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Equation System ◽  
Linear Differential Equations ◽  
Differential Equation System ◽  
Hurwitz Stability ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Stable Linear

For Hurwitz stable linear differential equation system with constant coefficients, we have proved continuity theorems which show how much change is permissible without disturbing the Hurwitz stability and the [Formula: see text]-Hurwitz stability. The results have been applied to the scalar–linear differential equations with order [Formula: see text] and some examples illustrating the efficiency of the theorems have been given.

scholarly journals ELEMENTARY METHOD OF SUBSTANTIATION AND IDEOLOGICAL MEANING OF THE THEORY OF THE SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS AND TRIGONOMETRIC FREE TERMS

2021 ◽  
pp. 105-114
Author(s):  
Zh. A. Sartabanov ◽  
A. Kh. Zhumagaziyev ◽  
A. A. Duyussova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Teaching Method ◽  
Second Order ◽  
School Mathematics ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Constant Coefficients ◽  
Linear Differential

In the article, adapted to the school course, the second order linear differential equations with constant coefficients and trigonometric free terms are investigated. The basic elementary methodological approaches to solving the equation are given. The solutions of the second order linear differential equation with constant coefficients and trigonometric free terms are investigated, which is a model of many phenomena. In addition, the applied values of the equation and its solutions were noted. The results obtained are presented in the form of theorems. The main novelty of the study is that these results are proved and generalized by elementary methods. These conclusions are proved in the framework of the methods of high school mathematics. This theory, known in general mathematics, is fully adapted to the implementation in secondary school mathematics and developed with the help of new elementary techniques that are understandable to the student. The main purpose of the research is to develop methods for solving a non-uniform linear differential equation of the second order with a constant coefficient at a level that a schoolboy can master. The result will be the creation of a special course program on the basics of ordinary differential equations in secondary schools of the natural-mathematical direction, the preparation of appropriate content material and providing them with a simple teaching method.

scholarly journals Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients

2017 ◽  
Vol 23 (3) ◽  
pp. 24-29
Author(s):  
Vasile Căruțașu
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Free Term ◽  
Order Linear ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential

Abstract For n-th order linear differential equations with constant coefficients, the problem to be solved is related to determining a particular solution, and then, with the general solution of n-th homogeneous linear differential equation with constant coefficients attached, to write the general solution of n-th linear differential equation with the given constant coefficients. In all the works that deal with this issue three situations are analyzed: the situation in which the free term is a polynomial P(x), the situation in which the free term is like P(x)· eα·x and lastly, the situation in which the free term is like eω·x · (P(x)· cos(β·x)+ Q(x)·sin(β·x)). In this study we aim to analyze if the free term is a combination of the three cases mentioned.

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

Studies in multiplicative solutions to linear differential equations

1987 ◽  
Vol 106 (3-4) ◽  
pp. 277-305 ◽  
Author(s):  
F. M. Arscott
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Floquet Theory ◽  
Special Functions ◽  
Linear Differential Equations ◽  
Ordinary Linear Differential Equation ◽  
Closed Path ◽  
Starting Point ◽  
Linear Differential

SynopsisGiven an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.

scholarly journals So-Called Cases of Failure in the Solution of Linear Differential Equations

1916 ◽  
Vol 8 (123) ◽  
pp. 258-262
Author(s):  
Eric H. Neville
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
General Equation ◽  
Linear Differential Equations ◽  
Effective Constant ◽  
Linear Differential ◽  
Made In

There are two ways in which the solution of a particular linear differential equation may “fail” although the solulion of a more general equation obtained by replacing certain constants by parameters is complete.where D as usual stands for d/dx.For the general equation(D — l)(D — m)y = enxthe perfectly general solution isA, B being independent arbitrary constants, but if we attempt to apply this solution to the particular equation (l), we find in the first place that the coincidence of n with l and m renders the first term infinite, and in the second place that the coincidence of m with l leaves us with only one effective constant, A + B. The method by which in the commoner textbooks the passage from the general solution to that of a particular equation is made in such cases as this is unconvincing.

New formula for solution of one class of linear differential equations of the second order with the variable coefficients

2020 ◽  
Vol 8 (3) ◽  
pp. 61-68
Author(s):  
Avyt Asanov ◽  
Kanykei Asanova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Linear Differential Equations ◽  
Common Solution ◽  
The Common ◽  
Linear Differential ◽  
New Formula

Exact solutions for linear and nonlinear differential equations play an important rolein theoretical and practical research. In particular many works have been devoted tofinding a formula for solving second order linear differential equations with variablecoefficients. In this paper we obtained the formula for the common solution of thelinear differential equation of the second order with the variable coefficients in themore common case. We also obtained the new formula for the solution of the Cauchyproblem for the linear differential equation of the second order with the variablecoefficients.Examples illustrating the application of the obtained formula for solvingsecond-order linear differential equations are given.Key words: The linear differential equation, the second order, the variablecoefficients,the new formula for the common solution, Cauchy problem, examples.

scholarly journals IV. On linear differential equations

1870 ◽  
Vol 18 (114-122) ◽  
pp. 118-119
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Rational Function ◽  
Linear Differential Equations ◽  
System Of Equations ◽  
Linear Differential

The condition that the linear differential equation (α + β x + γ x 2 ) d 2 u / dx 2 + (α' + β' x + γ' x 2 ) du / dx + (α'' + β" x + γ" x 2 ) u = 0 admits of an integral u = ϵ fφdx , where φ is a rational function of ( x ), is given by the system of equations

scholarly journals X. On linear differential equations.—No. V

1871 ◽  
Vol 19 (123-129) ◽  
pp. 526-528
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Linear Differential

Let us now endeavour to ascertain under what circumstance a linear differential equation admits a solution of the form P log e Q, where P and Q are rational functions of ( x ).

scholarly journals Pseudo-autonomous linear systems

1977 ◽  
Vol 16 (1) ◽  
pp. 61-65 ◽  
Author(s):  
W.A. Coppel
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Systems ◽  
Point Spectrum ◽  
Coefficient Matrix ◽  
Linear Differential Equations ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Linear Differential

Pseudo-autonomous linear differential equations are defined. A linear differential equation with bounded coefficient matrix is pseudo-autonomous if and only if it is almost reducible. A linear differential equation with recurrent coefficient matrix is pseudo-autonomous if and only if it has pure point spectrum.

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