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.

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.

scholarly journals Composition sum identities related to the distribution of coordinate values in a discrete simplex

10.37236/1498 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
R. Milson
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Algebraic Methods ◽  
Algebraic Proof ◽  
Geometric Properties ◽  
Linear Differential ◽  
Exponential Formula ◽  
Ordinary Linear Differential Equations ◽  
General Method

Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first and second order, ordinary, linear differential equations. Regarding the first class, the corresponding identities amount to a proof of the exponential formula of labelled counting. The identities in the second class can be used to establish certain geometric properties of the simplex of bounded, ordered, integer tuples. We present three theorems that support the conclusion that the inner dimensions of such an order simplex are, in a certain sense, more ample than the outer dimensions. As well, we give an algebraic proof of a bijection between two families of subsets in the order simplex, and inquire as to the possibility of establishing this bijection by combinatorial, rather than by algebraic methods.

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.

Periodic solutions of non-linear differential equations of the second order. V

1951 ◽  
Vol 47 (4) ◽  
pp. 752-755 ◽  
Author(s):  
Chike Obi
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

1·1. Let van der Pol's equation be taken in the formwhere ε1, ε2, k1 and k2 are small, and ω ≠ 0 is a constant, rational or irrational, independent of them.

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.

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