scholarly journals II. On the solution of linear differential equations

1848 ◽  
Vol 138 ◽  
pp. 31-54 ◽  
Keyword(s):  
Differential Equations ◽  
Differential Expression ◽  
Linear Differential Equations ◽  
Right Hand ◽  
Independent Variable ◽  
Linear Differential ◽  
The Right

If the operation of differentiation with regard to the independent variable x be denoted by the symbol D, and if ϕ (D) represent any function of D composed of integral powers positive or negative, or both positive and negative, it may easily be shown, that ϕ (D){ψ x. u } = ψ x. ϕ (D) u + ψ' x. ϕ' (D) u + ½ψ" x. ϕ" (D) u + 1/2.3 ψ"' x. ϕ"' (D) u + . . . (1.) and that ϕx .ψ(D) u = ψ(D){ ϕx. u } - ψ'(D){ ϕ'x. u } + ½ψ"(D){ ϕ"x. u } - 1/2.3ψ"'(D){ ϕ"'x. u } + . . (2.) and these general theorems are expressions of the laws under which the operations of differentiation, direct and inverse, combine with those operations which are de­noted by factors, functions of the independent variable. It will be perceived that the right-hand side of each of these equations is a linear differential expression; and whenever an expression assumes or can be made to assume either of these forms, its solution is determined; for the equations ϕ (D){ψ x. u } = P and ϕx . ψ(D) u = P are respectively equivalent to u = (ψ x ) -1 { ϕ (D)} -1 P and u = {ψ(D)} -1 (( ϕx ) -1 P).

Convergent Liouville–Green expansions for second-order linear differential equations, with an application to Bessel functions

1993 ◽  
Vol 440 (1908) ◽  
pp. 37-54 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Large Parameter ◽  
Order Linear ◽  
Factorial Series ◽  
Independent Variable ◽  
Linear Differential ◽  
Asymptotic Validity

A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville–Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re ( u ) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

Uniformly almost periodic solutions of non-linear differential equations of the second order I. General exposition

1955 ◽  
Vol 51 (4) ◽  
pp. 604-613
Author(s):  
Chike Obi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Almost Periodic ◽  
Non Linear ◽  
Independent Variable ◽  
Linear Differential ◽  
The Given ◽  
Analytical Expressions

1·1. A general problem in the theory of non-linear differential equations of the second order is: Given a non-linear differential equation of the second order uniformly almost periodic (u.a.p.) in the independent variable and with certain disposable constants (parameters), to find: (i) the non-trivial relations between these parameters such that the given differential equation has a non-periodic u.a.p. solution; (ii) the number of periodic and non-periodic u.a.p. solutions which correspond to each such relation; and (iii) explicit analytical expressions for the u.a.p. solutions when they exist.

Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 1

1985 ◽  
Vol 100 (3-4) ◽  
pp. 301-326 ◽  
Author(s):  
M. H. Lantsman
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Algebraic Equations ◽  
Order Of Growth ◽  
Constant Coefficients ◽  
Independent Variable ◽  
Linear Differential ◽  
Image Position ◽  
Asymptotic Representations

SynopsisWe consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equationcan be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

scholarly journals On the solution of linear differential equations

1851 ◽  
Vol 5 ◽  
pp. 937-939
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Similar Process ◽  
Distinct Process ◽  
Independent Variable ◽  
Linear Differential

The methods employed in this paper to effect the solution or reduction of linear differential equations consist of certain peculiar transformations, and each particular class of equations is transformed by a distinct process peculiarly its own. The reduction is effected by means of certain general theorems in the calculus of operations. The terms which form the first member of the first class of equations are functions of the symbols ɯ and τ, the latter being a function of x , and the former a function of x and D, x being the independent variable. This member of the equations contains two arbitrary functions of vs, and may therefore be of any order whatever. It likewise contains two simple factors, such for example as ɯ+ nk and which factors are taken away by the transformation employed, and consequently the equation is reduced an order lower; it is therefore integrated when of the second order. There is a series of equations of this class, each essentially distinct from the rest, yet all reducible by a similar process.

On the normal series satisfying linear differential equations.

1905 ◽  
Vol 74 (497-506) ◽  
pp. 339-340
Author(s):  
Emma Cunningham ◽  
Henry Frederick Baker
Keyword(s):  
Differential Equations ◽  
Simultaneous Equations ◽  
Linear Differential Equations ◽  
Normal Series ◽  
The Matrix ◽  
Independent Variable ◽  
Linear Differential ◽  
Square Matrix

If y denotes a column of n elements, and u is a square matrix of n rows and columns of elements, each of which is a function of the independent variable, n independent solutions of the system of simultaneous equations dy / dt = uy are given by the n columns of the matrix Ω( u ) = 1 + Q u + Q( u Q u ) + Q{ u Q ( u Q u )} + . . . ; where Q(Φ) denotes ᶴ t t 0 Φ dt .

scholarly journals A Relation between two Ordinary Linear Differential Equations of the Second Order

1935 ◽  
Vol 29 ◽  
pp. ix-xi
Author(s):  
D. G. Taylor
Keyword(s):  
Differential Equations ◽  
Order Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Independent Variable ◽  
Linear Differential ◽  
Image Position ◽  
Ordinary Linear Differential Equations

1. Between the solutions of the equationsa relation can be established, provided the functional symbols f, ø are inverse to one another. For example, let f (x) = sin x, then ø(ξ)= arcsin ξ, and the two equations areThe process consists in obtaining from (1) the second order equation satisfied by yx = dy/dx, making a change of independent variable in (2), and comparing the resulting equations.

scholarly journals XV. Invariants, convariants, and quotient-derivatives associated with linear differential equations

1888 ◽  
Vol 179 ◽  
pp. 377-489 ◽  
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Linear Character ◽  
Independent Variables ◽  
General Order ◽  
Independent Variable ◽  
The Subject ◽  
Linear Differential ◽  
Analytic Derivation

The present Memoir deals with a set of invariants and covariants of linear differential equations of general order. The set is proved to be complete, that is to say, every covariantive function of the same type can be expressed as a function of the members of the set, the only operations necessary for this expression being purely algebraical operations. The transformations, to which the differential equations are subjected, are supposed to be the most general consistent with the maintenance of their order and their linear character; they are, linear transformation of the dependent variable and arbitrary transformations of the independent variable. The covariantive property of the functions considered is constituted by the condition that, when the same functions are formed for the transformed equation, they are equal to the functions for the original equation, save as to a factor of the form ( dz / dx ) μ , where z and x are the two independent variables. The memoir, with the exception of a single and rather important digression, is occupied solely with investigations of the forms of the functions, of their interdependence, and of methods of construction. The earlier part deals chiefly with the synthetic derivation of the functions, the later part with their analytic derivation. Tables of the functions have not been calculated; in most cases the expressions of the functions are given in their forms as associated with the differential equation when it is taken in an implicitly general canonical form, and only in very few cases are functions given in connexion with an explicitly general form. Within these limits the subject of the memoir has been strictly confined; there is not, for instance, any attempt at classification of differential equations of the same order as discriminated by forms and values of invariants or covariants.

XIX. On the solution of linear differential equations

1851 ◽  
Vol 141 ◽  
pp. 461-482 ◽  
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Independent Variable ◽  
Linear Differential ◽  
General Method

If we consider the very different forms which the solutions of Differential Equations differing very little from each other frequently take, and the very different processes often required in each particular case to obtain the solution, we shall be led to conclude that the discovery of any universal or general method of solving them must be a hopeless case. We cannot therefore regard particular methods, especially when applicable to a large number of cases, as useless speculations. The present paper contains the solution of several classes of these equations effected by means of general theorems in the Calculus of Operations adapted to each particular class. For expla­nation of the symbols employed, let it be observed that D is put for and that d/dx , and that φ, λ, ψ, and X denote any functions of x , the independent variable, and are the same as φ( x ), λ( x ), &c.; and in like manner φ(D), λ(D), &c. will be used to denote the same functions of D.

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,∞).

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