8.—On a Class of Series Expansions in the Theory of Emden's Equation

1973 ◽  
Vol 71 (2) ◽  
pp. 95-110 ◽  
Author(s):  
Einar Hille
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Series Expansion ◽  
Laurent Series ◽  
Second Order ◽  
Series Expansions ◽  
Section 8 ◽  
Second Order Differential Equations ◽  
Bona Fide ◽  
Image Position

SynopsisThis paper deals with the nature of movable singularities of solutions of Emden's equationat which the solution becomes infinite. If m = 1 + 2/p with p > 1 an integer, then the solution becomes infinite at a given point x = c asBy the general theory of P. Painlevé on movable poles of solutions of non-linear second order differential equations this ‘pseudo-pole’ cannot actually be a pole of order p. Instead of a bona fide Laurent series at x = c we obtain a series expansion of the formwhere Pn(t) is a polynomial in t of degree at most [n/(2p + 2)]. The object of this paper is to derive these series and to prove convergence for p = 2. In this case deg [P6m] is strictly equal to m. For other values of p, see Section 8, Addenda.

Almost-conservative second-order differential equations

1975 ◽  
Vol 77 (1) ◽  
pp. 159-169 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Differential Equations ◽  
Rate Of Convergence ◽  
Second Order ◽  
Good Function ◽  
Second Order Differential Equations ◽  
Right Hand ◽  
Image Position ◽  
The Right

During the last thirty years an immense amount of research has been done on differential equations of the formwhere ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the formwhere the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assumeand for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

T-periodic solutions for some second order differential equations with singularities

1992 ◽  
Vol 120 (3-4) ◽  
pp. 231-243 ◽  
Author(s):  
Manuel del Pino ◽  
Raúl Manásevich ◽  
Alberto Montero
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Resonance Condition ◽  
Second Order ◽  
Fredholm Alternative ◽  
Positive Slope ◽  
Second Order Differential Equations ◽  
Image Position

SynopsisWe study the existence of T-periodic positive solutions of the equationwhere f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

1982 ◽  
Vol 25 (3) ◽  
pp. 291-295 ◽  
Author(s):  
Lance L. Littlejohn ◽  
Samuel D. Shore
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Image Position

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

scholarly journals New collocation integrator for solving dynamic problems

2020 ◽  
pp. 131-140
Author(s):  
V.A. Avdyshev ◽  
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Second Order ◽  
Dynamic Problems ◽  
Second Order Differential Equations ◽  
Mixed Systems

A new collocation integrator on Lobatto spacings is proposed for numerically solving mixed systems of first and second order differential equations of dynamic problems. The general theory of collocation integrators is described, from which the basic formulas of the new integrator are derived.

The Krall Polynomials as Solutions to a Second Order Differential Equation

1983 ◽  
Vol 26 (4) ◽  
pp. 410-417 ◽  
Author(s):  
Lance L. Littlejohn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Polynomial Solutions ◽  
Second Order Differential Equations ◽  
Image Position

AbstractA popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form1.1

The generalization of modal subspaces in the complex domain

1969 ◽  
Vol 65 (3) ◽  
pp. 731-740
Author(s):  
H. Swann ◽  
C. P. Atkinson ◽  
B. L. Dhoopar
Keyword(s):  
Positive Integer ◽  
Differential Equations ◽  
Real Variable ◽  
Complex Variables ◽  
Second Order ◽  
Complex Domain ◽  
The Real ◽  
Second Order Differential Equations ◽  
Non Linear ◽  
Image Position

The purpose of this paper is to present the concept of ‘modal subspaces’ for systems of coupled non-linear autonomous homogeneous second-order differential equations of the complex variables z1, z2. This development is an extension of the previous paper, entitled ‘Modal subspaces in the complex domain’, by Atkinson and Swann (6), which dealt with a pair of coupled non-linear autonomous homogeneous second-order differential equations of the formDifferentiation is with respect to a real variable t and ajk and bjk are constants which may be complex: n is any positive integer: f1 and f2 are the real and imaginary parts respectively of g1 and g2 are real and imaginary parts respectively of and zj = xj + iyj (j = 1, 2).

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.

Superexponentials: A Generalization of Hyperbolic and Trigonometric Functions

2017 ◽  
Vol 3 (1) ◽  
pp. 7
Author(s):  
Alfonso F. Agnew ◽  
Brandon Gentile ◽  
John H. Mathews
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Second Order ◽  
Series Expansions ◽  
Roots Of Unity ◽  
Trigonometric Functions ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Power Series Expansions

We construct and explore the properties of a generalization of hy- perbolic and trigonometric functions we cal l superexponentials. The general ization is based on the characteristic second-order differential equations (DE) these functions satisfy, and leads to functions satisfying analogous mth order equations and having many properties analogous to the usual hyperbolic and trigonometric functions. Roots of unity play a key role in providing the periodicity resulting in various properties. We also show how these functions solve the general initial value problem for the differential equations y(n) = y, and a look at the power series expansions reveal surprisingly simple patterns that clarify the properties of the superexponentials.

scholarly journals Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Willy Sarlet ◽  
Tom Mestdag
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Linear Transformation ◽  
Phase Synchronization ◽  
Second Order ◽  
Order System ◽  
Linear Transformations ◽  
Second Order Differential Equations ◽  
Geometric Context ◽  
The Given

<p style='text-indent:20px;'>The so-called method of phase synchronization has been advocated in a number of papers as a way of decoupling a system of linear second-order differential equations by a linear transformation of coordinates and velocities. This is a rather unusual approach because velocity-dependent transformations in general do not preserve the second-order character of differential equations. Moreover, at least in the case of linear transformations, such a velocity-dependent one defines by itself a second-order system, which need not have anything to do, in principle, with the given system or its reformulation. This aspect, and the related questions of compatibility it raises, seem to have been overlooked in the existing literature. The purpose of this paper is to clarify this issue and to suggest topics for further research in conjunction with the general theory of decoupling in a differential geometric context.</p>

Conjugacy of half-linear second-order differential equations

2000 ◽  
Vol 130 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Ondřej Došlý ◽  
Árpád Elbert
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Focal Point ◽  
Second Order ◽  
Oscillation Criterion ◽  
Riccati Transformation ◽  
Second Order Differential Equations ◽  
Generalized Riccati Transformation ◽  
Image Position

Focal point and conjugacy criteria for the half-linear second-order differential equation are obtained using the generalized Riccati transformation. An oscillation criterion is given in case when the function c(t) is periodic.

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