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

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.

scholarly journals The distribution of zeros of solutions for a class of third order differential equation

2021 ◽  
Vol 40 (5) ◽  
pp. 1301-1321
Author(s):  
Clemente Cesarano ◽  
Mohammed A. Arahet ◽  
Tareq M. Al-Shami
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Lower Bounds ◽  
Order Differential Equation ◽  
Linear Differential Equations ◽  
Distribution Of Zeros ◽  
Third Order ◽  
Order Linear ◽  
Zeros Of Solutions ◽  
Linear Differential

For third order linear differential equations of the form r(t)x'(t)''+ p(t)x'(t) + q(t)x(t) = 0; we will establish lower bounds for the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardyís inequality, some generalizations of Opialís inequality and Boydís inequality.

Hille–Wintner type comparison criteria for the half-linear differential equations of third order

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Jozef Kiseľák
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Linear Differential Equations ◽  
Third Order ◽  
Weak Property ◽  
Comparison Criteria ◽  
Linear Differential

AbstractWe obtain an analogue of the integral Hille–Wintner comparison theorem for the half-linear differential equations of third order. We also give an example involving a differential equation of Euler type, which gives a condition under which half-linear differential equations have weak property

scholarly journals Some Extensions of Pincherle's Polynomials

1920 ◽  
Vol 39 ◽  
pp. 21-24 ◽  
Author(s):  
Pierre Humbert
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Third Order ◽  
The Third ◽  
Hypergeometric Type ◽  
Linear Differential ◽  
Image Position

The polynomials which satisfy linear differential equations of the second order and of the hypergeometric type have been the object of extensive work, and very few properties of them remain now hidden; the student who seeks in that direction a subject for research is compelled to look, not after these functions themselves but after generalisations of them. Among these may be set in first place the polynomials connected with a differential equation of the third order and of the extended hypergeometric type, of which a general theory has been given by Goursat. The number of such polynomials of which properties have been studied in particular is rather small; in fact, Appell's polynomialsand Pincherle's polynomials, arising from the expansionsare, so far as I know, the only well-known ones. To show what can be done in these ways, I shall briefly give the definition and principal properties of some polynomials analogous to Pincherle's and of some allied functions.

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 non-homogeneous canonical third-order linear differential equations

1994 ◽  
Vol 57 (2) ◽  
pp. 138-148 ◽  
Author(s):  
N. Parhi
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Linear Differential Equations ◽  
Third Order ◽  
Order Linear ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

AbstractIn this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

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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