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 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.

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 Renormalization, Isogenies, and Rational Symmetries of Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-44 ◽  
Author(s):  
A. Bostan ◽  
S. Boukraa ◽  
S. Hassani ◽  
J.-M. Maillard ◽  
J.-A. Weil ◽  
...  
Keyword(s):  
Differential Equation ◽  
Renormalization Group ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Exact Representation ◽  
Symmetries Of Differential Equations ◽  
Linear Differential

We give an example of infinite-order rational transformation that leaves a linear differential equation covariant. This example can be seen as a nontrivial but still simple illustration of an exact representation of the renormalization group.

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 Growth of solutions of second order complex linear differential equations with entire coefficients

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 275-284 ◽  
Author(s):  
Jianren Long
Keyword(s):  
Entire Function ◽  
Differential Equations ◽  
Nontrivial Solution ◽  
Infinite Order ◽  
Entire Functions ◽  
Order Complex ◽  
Complex Linear ◽  
Linear Differential ◽  
The One ◽  
Growth Of Solutions

Some new conditions on the entire coefficients A(z) and B(z), which guarantee every nontrivial solution of f''+A(z) f'+B(z) f = 0 is of infinite order, are given in this paper. Two classes of entire functions are involved in these conditions, the one is entire functions having Fabry gaps, the another is function extremal for Yang?s inequality. Moreover, a kind of entire function having finite Borel exception value is considered.

scholarly journals On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Extremal Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianren Long ◽  
Chunhui Qiu ◽  
Pengcheng Wu
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Infinite Order ◽  
Entire Functions ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We consider that the linear differential equationsf(k)+Ak-1(z)f(k-1)+⋯+A1(z)f′+A0(z)f=0, whereAj  (j=0,1,…,k-1), are entire functions. Assume that there existsl∈{1,2,…,k-1}, such thatAlis extremal forYang'sinequality; then we will give some conditions on other coefficients which can guarantee that every solutionf(≢0)of the equation is of infinite order. More specifically, we estimate the lower bound of hyperorder offif every solutionf(≢0)of the equation is of infinite order.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals Growth of solutions of second order linear differential equations with extremal functions for Denjoy's conjecture as coefficients

2016 ◽  
Vol 47 (2) ◽  
pp. 237-247 ◽  
Author(s):  
Jianren Long
Keyword(s):  
Differential Equations ◽  
Infinite Order ◽  
Classical Problem ◽  
Extremal Functions ◽  
Nontrivial Solutions ◽  
Deficient Value ◽  
Borel Exceptional Value ◽  
Exceptional Value ◽  
Linear Differential ◽  
Growth Of Solutions

The classical problem of finding conditions on the entire coefficients $A(z)$ and $B(z)$ guaranteeing that all nontrivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order is discussed. Some such conditions which involve deficient value, Borel exceptional value and extremal functions for Denjoy's conjecture are obtained.

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