The Existence of Continuable Solutions of a Second Order Differential Equation

1977 ◽  
Vol 29 (3) ◽  
pp. 472-479 ◽  
Author(s):  
G. J. Butler
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Real Line ◽  
Order Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
The Real ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Sign Restriction

A much-studied equation in recent years has been the second order nonlinear ordinary differential equationwhere q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

Oscillation Criteria For Second Order Superlinear Differential Equations

1989 ◽  
Vol 41 (2) ◽  
pp. 321-340 ◽  
Author(s):  
CH. G. Philos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Continuous Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Line ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
Oscillation Criteria ◽  
Image Position

This paper is concerned with the question of oscillation of the solutions of second order superlinear ordinary differential equations with alternating coefficients.Consider the second order nonlinear ordinary differential equationwhere a is a continuous function on the interval [t0, ∞), t0 > 0, and / is a continuous function on the real line R, which is continuously differentia t e , except possibly at 0, and satisfies.

On the Forced Lienard Equation

1969 ◽  
Vol 12 (1) ◽  
pp. 79-84 ◽  
Author(s):  
R.R. Stevens
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Liénard Equation ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Lienard Equation

We consider the second order differential equation(1)with the assumptions that(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.Also, throughout this paper, F(x) denotes an antiderivative of f(x).

scholarly journals Dirichlet problem for a second-order ordinary differential equation with a distributed differentiation operator

2021 ◽  
Vol 21 (4) ◽  
pp. 37-44
Author(s):  
B.I. Efendiev ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Dirichlet Problem ◽  
Function Method ◽  
Order Differential Equation ◽  
Variable Coefficients ◽  
Differentiation Operator ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation

For an ordinary second-order differential equation with an operator of continuously distributed differentiation with variable coefficients, a solution to the Dirichlet problem is constructed using the Green’s function method.

scholarly journals Bounded Solutions of the Second Order Differential Equation x ?+f(x) x ?+g(x)=u(t)

2015 ◽  
Vol 12 (4) ◽  
pp. 822-825
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Bounded Solutions ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation ◽  
Derivatives Of

In this paper we prove the boundedness of the solutions and their derivatives of the second order ordinary differential equation x ?+f(x) x ?+g(x)=u(t), under certain conditions on f,g and u. Our results are generalization of those given in [1].

scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

Periodic solutions for a second-order differential equation with indefinite weak singularity

2019 ◽  
Vol 149 (5) ◽  
pp. 1135-1152 ◽  
Author(s):  
José Godoy ◽  
Manuel Zamora
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Second Order ◽  
Constant Sign ◽  
Piecewise Constant ◽  
Weak Singularity ◽  
Second Order Differential Equation ◽  
Image Position

AbstractAs a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation $${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation

1949 ◽  
Vol 1 (2) ◽  
pp. 191-198 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Arbitrary Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Partial Differential ◽  
Second Order Differential Equation ◽  
Image Position

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

Cauchy problem for second-order ordinary differential equation with continuously distributed differentiation operator

2020 ◽  
Vol 20 (1) ◽  
pp. 15-20
Author(s):  
B.I. Efendiev ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Order Differential Equation ◽  
Differentiation Operator ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation ◽  
The Cauchy Problem

In this paper, we construct the fundamental solution for ordinary second-order differential equation with continuously distributed differentiation operator. With the help of fundamental solution the solution of the Cauchy problem is written out.

On the stability of a second-order differential equation

1965 ◽  
Vol 5 (1) ◽  
pp. 8-16 ◽  
Author(s):  
D. E. Daykin ◽  
K. W. Chang
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
The Stability ◽  
Image Position ◽  
Familiar Form

In this note we discuss the stability at the origin of the solutions of the differential equation where a dot indicates a differentiation with respect to time, and α, β are real-valued functions of any arguments. We tacitly assume that α, β are such that solutions to (1) do in fact exist. Under the transformation equation (1) takes the equivalent familiar form .

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