scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

On a Linear Partial Differential Equation of Hyperbolic Type

1922 ◽  
Vol 41 ◽  
pp. 76-81
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Order Partial Differential Equation ◽  
Sound Waves ◽  
Partial Differential ◽  
Method Of Solution ◽  
Image Position

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equationwhere α, β, γ are functions of x and y.

scholarly journals Exceptional Integrals of a not completely Integrable Total Differential Equation

1953 ◽  
Vol 1 (3) ◽  
pp. 137-138
Author(s):  
H. T. H. Piaggio
Keyword(s):  
Differential Equation ◽  
Completely Integrable ◽  
Total Differential ◽  
Image Position ◽  
De La Vallée Poussin

1. There are exceptional integrals of the total differential equationin the case when it is not completely integrable, and so when the invariantis not identically zero, which do not seem to be mentioned by any standard authorities such as Cartan, Goursat, de la Vallée Poussin, and Schouten and Kulk. These are integrals of (1) which do not reduce I to zero. They arise only when the first partial derivates of P, Q, R are not all continuous. A simple example is z = 0 as an integral of

Analytical theory of non-linear oscillations

1974 ◽  
Vol 76 (1) ◽  
pp. 285-296 ◽  
Author(s):  
Chike Obi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Existence Theorem ◽  
Wide Class ◽  
Periodic Oscillations ◽  
Real Domain ◽  
Non Linear ◽  
General Existence ◽  
Linear Differential ◽  
Image Position

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equationwhere g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

scholarly journals A comparison theorem and the forced oscillation

1975 ◽  
Vol 13 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Present Author ◽  
Forced Oscillation ◽  
Oscillatory Behavior ◽  
The Subject ◽  
Image Position ◽  
Delay Differential ◽  
Oscillation Theorems

Consider the n-th order delay differential equationIn the last few years, the oscillatory behavior of delay differential equations has been the subject of intensive investigations. But much less is known about the equation (A) with small forcing term q(t). The only papers devoted to this problem are by Kartsatos, Kusano, and the present author. The purpose of this paper is to prove some new oscillation theorems which contain the previous results.

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

scholarly journals On Runge-Kutta processes of high order

1964 ◽  
Vol 4 (2) ◽  
pp. 179-194 ◽  
Author(s):  
J. C. Butcher
Keyword(s):  
Differential Equation ◽  
High Order ◽  
Runge Kutta ◽  
Image Position

An (explicit) Runge-Kutta process is a means of numerically solving the differential equation , at the point x = x0+h, where y, f may be vectors.

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