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 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 the Systems of Linear Differential Equations with Constant Coefficients

2017 ◽  
Vol 23 (3) ◽  
pp. 30-36
Author(s):  
Vasile Căruțașu
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Homogeneous System ◽  
Linear Differential Equations ◽  
Order Linear ◽  
System A ◽  
Variation Of Constants ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential

Abstract As with the n-th order linear differential equations with constant coefficients, the problem to be solved is related to determining a particular solution, and then, using the general solution of the attached homogeneous system of linear differential equations with constant coefficients, to write the general solution of the initially given system. For homogeneous systems of linear differential equations with constant coefficients, the determination of the general solution is the method of eliminating or reducing which make the system a linear differential equation of the same order as that of the system, and its methods of solving it applies or the method of own values and vectors. If the system is non-homogeneous, then we also have to determine a particular solution that can be done in the same way as in the case of n-th order differential equations with constant coefficients, if the method of reduction or elimination was used, or the method of variation of constants, regardless of the method used to determine the general solution of the attached homogenous system of linear differential equations with constant coefficients. Whichever method is used, determining a particular solution for a system of linear differential equations with constant coefficients is difficult, in this study being proposed a method similar to that of n-th order linear differential equations with constant coefficients.

scholarly journals Determining a Particular Solution for N-th Order Linear Differential Equations with Constant Coefficients that Have the Free Term in the Following Form ea·x·P(x) ·cosnx or eb·x·Q(x)·sinnx

2020 ◽  
Vol 26 (3) ◽  
pp. 49-54
Author(s):  
Vasile Căruţaşu ◽  
Alexandru Hampu
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
The Other ◽  
Free Term ◽  
Order Linear ◽  
Other Hand ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential ◽  
The Way

AbstractA particular solution for the n-th order linear differential equations with constant coefficients that are free of term such as P(x)· ea·x ·cosnx or/and Q(x)·eb·x· sinnx, n∈ N, can be determined based on two elements: the way in which cosnx and sinnx can develop, and, on the other hand on the way a particular solution for the free terms P(x) ·ea·x·cosnx or/and Q(x)·eb·x·sinmx, n ∈N is sought. We can, of course, write the way a particular solution looks also in the case we have a combination of the two terms or more terms of this kind.

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.

scholarly journals Oscillation of first order linear differential equations with several non-monotone delays

2018 ◽  
Vol 16 (1) ◽  
pp. 83-94
Author(s):  
E.R. Attia ◽  
V. Benekas ◽  
H.A. El-Morshedy ◽  
I.P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
First Order ◽  
Linear Differential

AbstractConsider the first-order linear differential equation with several retarded arguments$$\begin{array}{} \displaystyle x^{\prime }(t)+\sum\limits_{k=1}^{n}p_{k}(t)x(\tau _{k}(t))=0,\;\;\;t\geq t_{0}, \end{array} $$where the functions pk, τk ∈ C([t0, ∞), ℝ+), τk(t) < t for t ≥ t0 and limt→∞τk(t) = ∞, for every k = 1, 2, …, n. Oscillation conditions which essentially improve known results in the literature are established. An example illustrating the results is given.

scholarly journals Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
A. Javadian ◽  
E. Sorouri ◽  
G. H. Kim ◽  
M. Eshaghi Gordji
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Open Interval ◽  
Second Order ◽  
Linear Differential Equations ◽  
Ulam Stability ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form , with condition that there exists a nonzero in such that and is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known differential equations.

SOME PROPERTIES OF COMBINATION OF SOLUTIONS TO SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ANALYTIC COEFFICIENTS OF [P;Q]-ORDER IN THE UNIT DISC

2015 ◽  
Vol 1 (1) ◽  
pp. 11-18
Author(s):  
Benharrat Belaïdi ◽  
Zinelâabidine Latreuch
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Unit Disc ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linear Differential

In this paper, we consider some properties on the growth and oscillation of combination of solutions of the linear differential equation \[f'' + A(z) f' + B (z) f = 0\] with analytic coefficients A(z) and B (z) with [p; q]-order in the unit disc $\Delta = \{z \in \mathbb{C} : |z| < 1\}$.

scholarly journals Analytic solution of transcendental equations

2010 ◽  
Vol 20 (4) ◽  
pp. 671-677 ◽  
Author(s):  
Henryk Górecki
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Analytic Solution ◽  
Linear Differential Equations ◽  
Order Linear Differential Equation ◽  
Decomposition Technique ◽  
Order Linear ◽  
Transcendental Equations ◽  
Linear Differential

Analytic solution of transcendental equationsA decomposition technique of the solution of ann-th order linear differential equation into a set of solutions of 2-nd order linear differential equations is presented.

scholarly journals Some Properties of Solutions of Second-Order Linear Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Zinelaâbidine Latreuch ◽  
Benharrat Belaïdi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Finite Order ◽  
Entire Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linearly Independent ◽  
Linear Differential

We study the growth and oscillation of gf=d1f1+d2f2, where d1 and d2 are entire functions of finite order not all vanishing identically and f1 and f2 are two linearly independent solutions of the linear differential equation f′′+A(z)f=0.

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.

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