The function and associated polynomials

1951 ◽  
Vol 47 (2) ◽  
pp. 309-314 ◽  
Author(s):  
D. F. Lawden
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Linear Difference Equations ◽  
Transform Method ◽  
Associated Polynomials ◽  
The Laplace Transform ◽  
Linear Differential ◽  
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A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation

Solution Space Decompositions for nth Order Linear Differential Equations

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 ◽  
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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).

scholarly journals Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

2018 ◽  
Vol 23 (4) ◽  
pp. 76
Author(s):  
Julia Gregori ◽  
Juan López ◽  
Marc Sanz
Keyword(s):  
Differential Equations ◽  
Homogeneous Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mean Square ◽  
Recent Contribution ◽  
Transform Method ◽  
Order Linear ◽  
Random Power Series ◽  
Linear Differential

The objective of this paper is to complete certain issues from our recent contribution (Calatayud, J.; Cortés, J.-C.; Jornet, M.; Villafuerte, L. Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Adv. Differ. Equ. 2018, 392, 1–29, doi:10.1186/s13662-018-1848-8). We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via a habitual Lipschitz condition that extends the classical Picard theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, the latter being a reformulation of our random Fröbenius method.

A proof of characterisation of oscillation for higher-order neutral differential equations of mixed type by the Laplace transform

1994 ◽  
Vol 124 (5) ◽  
pp. 909-916 ◽  
Author(s):  
O. Arino ◽  
M. A. El Attar
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
General Expression ◽  
Exponential Growth ◽  
Mixed Type ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Equations Of Mixed Type ◽  
The Laplace Transform ◽  
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Consider the general expression of such equations in the formwhere Ai, Bj, ∊ ℝ, δo = 0 dn/ 0, dn are n-derivatives, n ≧ l, the σj'S and δj,'s respectively, are ordered as an increasing family with possibly positive and negative terms. These are the deviating arguments. In this paper, we provide a proof of this result based on the use of the Laplace transform. Our method involves new results regarding the exponential growth of positive solutions for such equations.

scholarly journals The Elliptic Cylinder Functions of the Second Kind

1914 ◽  
Vol 33 ◽  
pp. 2-13 ◽  
Author(s):  
E. Lindsay Ince
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Elliptic Cylinder ◽  
Linear Differential Equations ◽  
Periodic Functions ◽  
Linear Differential ◽  
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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.

Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
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SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

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