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.

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 On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

scholarly journals III. On linear differential equations.— No. II

1870 ◽  
Vol 18 (114-122) ◽  
pp. 210-212
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
General Linear ◽  
Linear Differential

The principles laid down in my former paper will enable us to integrate a proposed differential equation, when the solution can be expressed in the form—P/Q, where P, Q, to are rational and entire functions of ( x ). Let (α 0 +α 1 x +α 2 x 2 + ... +α m x m ) d n y / dx n + (β 0 +β 1 +β 2 x 2 + ... +α m x m ) d n-1 y / dx n-1 + (γ 0 +γ 1 x +γ 2 x 2 + ... +γ m x m ) d n-2 y / dx n-2 +.... +(λ 0 +λ 1 +λ 2 x 2 + ... +λ m x m ) y =0 be the general linear differential equation of the nth order, where none of the indices of ( x ) in the coefficients of the succeeding terms are greater than those in the coefficients of the two first.

scholarly journals Some Oscillation Results of Higher-Order Linear Differential Equations with Meromorphic Coefficients

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

We investigate the growth of solutions of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. We also discuss the relationship between small functions and solutions of such equations.

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.

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

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