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 ◽  
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).

16.—Square Integrable Solutions of Perturbed Linear Differential Equations

1975 ◽  
Vol 73 ◽  
pp. 251-254 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Differential Expression ◽  
Function Space ◽  
Hilbert Function ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Image Position ◽  
Hilbert Function Space

SynopsisThis paper is concerned with solutions of the ordinary differential equationwhere ℒ is a real formally self-adjoint, linear differential expression of order 2n, and the perturbed term f satisfiesfor some σ∈[0, 1]. Here λ(·) is locally integrable on [0,∞).In particular it is shown, under circumstances detailed in the text, that (*) possesses solutions in the Hilbert function space L2(0,∞).

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 ◽  
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.

scholarly journals On the non-existence of L2 solutions of a class of non-linear differential equations

1965 ◽  
Vol 14 (4) ◽  
pp. 257-268 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Half Line ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equationdoes not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

scholarly journals info geting lost in plates 7u,8,9,11 and 13 - I. On the normal series satisfying linear differential equations

1906 ◽  
Vol 205 (387-401) ◽  
pp. 1-35 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
London Mathematical Society ◽  
Linear Ordinary Differential Equation ◽  
Linear Differential Equations ◽  
Independent Variables ◽  
Normal Series ◽  
The Matrix ◽  
Linear Differential

1. The present paper is suggested by that of Dr. H. F. Baker in the ‘Proceedings of the London Mathematical Society,’ vol. xxxv., p. 333, “On the Integration of Linear Differential Equations.” In that paper a linear ordinary differential equation of order n is considered as derived from a system of n linear simultaneous differential equations dx i / dt = u i1 x +.....+ u i n x n ( i = 1... n ), or, in abbreviated notation, dx / dt = ux , where u is a square matrix of n rows and columns whose elements are functions of t , and x denotes a column of n independent variables. A symbolic solution of this system is there given and denoted by the symbol Ω( u ). This is a matrix of n rows and columns formed from u as follows :—Q ( ϕ ) is the matrix of which each element is the t -integral from t 0 to t of the corresponding element of ϕ , ϕ being any matrix of n rows and columns; then Ω( u ) = 1+Q u +Q u Q u +Q u Q u Q u ..... ad inf ., where the operator Q affects the whole of the part following it in any term.

scholarly journals Focal decompositions for linear differential equations of the second order

2003 ◽  
Vol 2003 (14) ◽  
pp. 813-821 ◽  
Author(s):  
L. Birbrair ◽  
M. Sobolevsky ◽  
P. Sobolevskii
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Equivalence Relation ◽  
Complete Classification ◽  
Second Order ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Second Order Equations

Focal decomposition associated to an ordinary differential equation of the second order is a partition of the set of all two-points boundary value problems according to the number of their solutions. Two equations are called focally equivalent if there exists a homomorphism of the set of two-points problems to itself such that the image of the focal decomposition associated to the first equation is a focal decomposition associated to the second one. In this paper, we present a complete classification for linear second-order equations with respect to this equivalence relation.

On the relation between distinct particular solutions of equation

1993 ◽  
Vol 123 (5) ◽  
pp. 917-926
Author(s):  
Guan Ke-ying ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Particular Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThere exists a relation (1.5) between any n + 2 distinct particular solutions of the differential equationIn this paper, we show that when and only when n = 0, 1 and 2, this relation can be represented by the following form:provided the form of this relation function Φn depends only on n and is independent of the coefficients of the equation. This result reveals interesting properties of these non-linear differential equations.

scholarly journals Inequalities for solutions of linear differential equations a contribution to the theory of servomechanisms

1952 ◽  
Vol 38 ◽  
pp. 13-16 ◽  
Author(s):  
Hans Bückner
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Real Function ◽  
Order Differential Equation ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Image Position ◽  
Imaginary Roots

Consider the nth order differential equationwhere the coefficients cv are real constants and f is a real function continuous in the interval a≦ x ≦ b. The following theorem will be proved in §4:If the characteristic equation of (I) has no purely imaginary roots, then a particular integral η (x) can always be found which satisfies the inequalitywhere C is a certain function of the cv only and M is the maximum of |f|. In particular we may take C = 1 if all roots of the characteristic equation are real.

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.

Klein's Oscillation Theorem for Periodic Boundary Conditions

1971 ◽  
Vol 23 (4) ◽  
pp. 699-703 ◽  
Author(s):  
A. Howe
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Periodic Boundary Conditions ◽  
Linear Differential Equations ◽  
Oscillation Theorem ◽  
Linear Differential ◽  
Image Position

Multiparameter eigenvalue problems for systems of linear differential equations with homogeneous boundary conditions have been considered by Ince [4] and Richardson [5, 6], and more recently Faierman [3] has considered their completeness and expansion theorems. A survey of eigenvalue problems with several parameters, in mathematics, is given by Atkinson [1].We consider the two differential equations:1a1bwhere p1’(x), q1(x), A1(x), B1(x) and p2’(y), q2(y), A2(y), B2(y) are continuous for x ∈ [a1, b1] and y ∈ [a2, b2] respectively, and p1 (x) > 0(x ∈ [a1, b1]), p2(y) > 0 (y ∈ [a2, b2]), p1(a1) = p1(b1), p2(a2) = p2(b2). The differential equations (1) will be subjected to the periodic boundary conditions.2a2bLet us consider a single differential equation

A boundedness theorem for non-linear differential equations of the second order

1951 ◽  
Vol 47 (1) ◽  
pp. 49-54 ◽  
Author(s):  
G. E. H. Reuter
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Damping Factor ◽  
Force Term ◽  
Restoring Force ◽  
Boundedness Theorem ◽  
Linear Differential ◽  
Image Position

1. This paper deals with the differential equation(dots denoting derivatives with respect to t), where for large x the ‘restoring force’ term g(x) has the sign of x and the ‘damping factor’ kf(x) is positive on the average. It will be shown that every solution of (1) ultimately (for sufficiently large t) satisfieswith B independent of k. The conditions on f(x), g(x) and p(t) (stated in §§ 2, 3) are rather milder than those assumed by Cartwright and Littlewood (1, 2) and Newman (3) in proving similar results.

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