Admissible pair of spaces for non-correctly solvable linear differential equations

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

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

Filomat ◽

10.2298/fil1913013l ◽

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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Calculating Galois groups of third-order linear differential equations with parameters

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500389 ◽

2018 ◽

Vol 20 (04) ◽

pp. 1750038

Author(s):

Andrei Minchenko ◽

Alexey Ovchinnikov

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Galois Group ◽

Galois Groups ◽

Linear Differential Equations ◽

Unipotent Radical ◽

Third Order ◽

Differential Galois Group ◽

Bessel Differential Equation ◽

Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

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Solution Space Decompositions for nth Order Linear Differential Equations

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-062-x ◽

1975 ◽

Vol 27 (3) ◽

pp. 508-512

Author(s):

G. B. Gustafson ◽

S. Sedziwy

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Differential Equations ◽

Solution Space ◽

Decomposition Theorem ◽

Linear Differential Equations ◽

Direct Sum ◽

Direct Sum Decomposition ◽

Linear Differential ◽

Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

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On the Nature of the Spectrum of Singular Second Order Linear Differential Equations

Canadian Journal of Mathematics ◽

10.4153/cjm-1951-039-0 ◽

1951 ◽

Vol 3 ◽

pp. 335-338 ◽

Cited By ~ 7

Author(s):

E. A. Coddington ◽

N. Levinson

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Continuous Functions ◽

Second Order ◽

Real Parameter ◽

Linear Differential Equations ◽

Order Linear ◽

The Real ◽

Linear Differential

Let p(x) > 0, q(x) be two real-valued continuous functions on . Suppose that the differential equation with the real parameter λ

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

Abstract and Applied Analysis ◽

10.1155/2012/451825 ◽

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.

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Existence of conjugate points for self-adjoint linear differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023969 ◽

1989 ◽

Vol 113 (1-2) ◽

pp. 73-85 ◽

Cited By ~ 3

Author(s):

Ondřej Došlý

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Linear Differential Equations ◽

Conjugate Points ◽

General Differential Equation ◽

Linear Differential

SynopsisThe conjecture of Muller-Pfeiffer [4] concerning the oscillation behaviour of the differential equation (–l)n(p(x)y(n))(n) + q(x)y = 0 is proved, and a similar conjecture concerning the more general differential equation ∑nk=0(−l)k(Pk(x)y(k)(k + q(x)y= 0 is formulated.

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Studies in multiplicative solutions to linear differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500018424 ◽

1987 ◽

Vol 106 (3-4) ◽

pp. 277-305 ◽

Cited By ~ 2

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.

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The Elliptic Cylinder Functions of the Second Kind

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500002285 ◽

1914 ◽

Vol 33 ◽

pp. 2-13 ◽

Cited By ~ 4

Author(s):

E. Lindsay Ince

Keyword(s):

Differential Equation ◽

General Solution ◽

Differential Equations ◽

Elliptic Cylinder ◽

Linear Differential Equations ◽

Periodic Functions ◽

Linear Differential ◽

Image Position

The differential equation of Mathieu, or the equation of the elliptic cylinder functionsis known by the theory of linear differential equations to have a general solution of the typeφ and ψ being periodic functions of z, with period 2π.

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