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.

V. Mechanical integration of the linear differential equations of the second order with variable coefficients

1876 ◽  
Vol 24 (164-170) ◽  
pp. 269-271 ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Second Order ◽  
Sectional Area ◽  
Practical Solution ◽  
Hanging Chain ◽  
Linear Differential

Every linear differential equation of the second order may, as is known, be reduced to the form d / dx (1/P du / dx ) = u , . . . . . . (1) where P is any given function of x . On account of the great importance of this equation in mathematical physics (vibrations of a non-uniform stretched cord, of a hanging chain, water in a canal of non-uniform breadth and depth, of air in a pipe of non-uniform sectional area, conduction of heat along a bar of non-uniform fiction or non-uniform conductivity, Laplace’s differential equation of the tides, &c. &c.), I have long endeavoured to obtain a means of faciliiting its practical solution.

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 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 Two-Point Oscillation for a Class of Second-Order Damped Linear Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Kong Xiang-Cong ◽  
Zheng Zhao-Wen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Comparison Theorem ◽  
Second Order ◽  
Linear Differential Equations ◽  
Damping Term ◽  
Linear Differential

Using the comparison theorem, the two-point oscillation for linear differential equation with damping term is considered, where . Results are obtained that or imply the two-point oscillation of the equation.

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.

scholarly journals IV. On linear differential equations.—No. VII

1873 ◽  
Vol 21 (139-147) ◽  
pp. 20-21
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Similar Investigation ◽  
Linear Differential

I am desirous to conclude this series of papers with some remarks on the solutions of differential equations considered as transcendents. I shall take the linear differential equation of the second order, ( α + βx + γx 2 ) d 2 y / dx 2 +( α' + β'x + γ'x 2 ) dy / dx + ( α'' + β'' + γ''x 2 ) y = 0, which will be sufficient, as it will be seen at once that similar investigation apply generally.

scholarly journals Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 235
Author(s):  
Zuzana Pátíková ◽  
Simona Fišnarová
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Type Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Type Criterion ◽  
Linear Differential

We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille–Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

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.

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