A Geometrical Approach to the Second-Order Linear Differential Equation

1962 ◽  
Vol 14 ◽  
pp. 349-358 ◽  
Author(s):  
C. M. Petty ◽  
J. E. Barry
Keyword(s):  
Differential Equation ◽  
Minkowski Plane ◽  
Geometrical Approach ◽  
Order Linear Differential Equation ◽  
The Real ◽  
Additional Tool ◽  
Real Solutions ◽  
Convex Curves ◽  
Linear Differential ◽  
Image Position

In this paper various concepts intrinsically defined by the differential equation1.1are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).Throughout this paper, R(t) is a real-valued, continuous function of t on the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.

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

Linear differential equations with constant coefficients

2019 ◽  
Vol 103 (557) ◽  
pp. 257-264
Author(s):  
Bethany Fralick ◽  
Reginald Koo
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Auxiliary Equation ◽  
Real Solutions ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Complex Solutions ◽  
Image Position ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

We consider the second order homogeneous linear differential equation (H) $${ ay'' + by' + cy = 0 }$$ with real coefficients a, b, c, and a ≠ 0. The function y = emx is a solution if, and only if, m satisfies the auxiliary equation am2 + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e(p ± iq)x are complex solutions of (H). Nevertheless, real solutions are given by y = c1epx cos qx + c2epx sin qx.

scholarly journals ON LAGUERRE–SOBOLEV TYPE ORTHOGONAL POLYNOMIALS: ZEROS AND ELECTROSTATIC INTERPRETATION

2013 ◽  
Vol 55 (1) ◽  
pp. 39-54
Author(s):  
LUIS ALEJANDRO MOLANO MOLANO
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Inner Product ◽  
Sobolev Type ◽  
Electrostatic Model ◽  
Order Linear Differential Equation ◽  
Standard Methods ◽  
Linear Differential ◽  
Electrostatic Interpretation ◽  
Image Position

AbstractWe study the sequence of monic polynomials orthogonal with respect to inner product $$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$ where $\alpha \gt - 1$, $M\geq 0$, $N\geq 0$, $\zeta \lt 0$, and $p$ and $q$ are polynomials with real coefficients. We deduce some interlacing properties of their zeros and, by using standard methods, we find a second-order linear differential equation satisfied by the polynomials and discuss an electrostatic model of their zeros.

Stability and Response of Systems Satisfying a Second-Order Linear Differential Equation with Time-Dependent Coefficients

1957 ◽  
Vol 8 (1) ◽  
pp. 78-86
Author(s):  
A. W. Babister
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Linear Differential Equation ◽  
Second Order ◽  
Time Dependent ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Oscillatory Nature ◽  
Linear Differential ◽  
Image Position

SummaryThe differential equation considered iswhere all the a’s and b’s are real constants.The nature of the solution is investigated in the neighbourhood of the singular point and the conditions are found for logarithmic terms to be absent.The conditions for stability for large values of τ are determined; the system is stable ifare all positive for large values of τ.The form of the response is considered and its oscillatory (or non-oscillatory) nature investigated. The Sonin-Polya theorem is used to determine simple inequalities which must hold between the coefficients of the differential equation in any interval for the relative maxima of | x | to form an increasing or decreasing sequence in that interval.

scholarly journals A stability result for the linear differential equation x”+f(t)x = 0

1967 ◽  
Vol 7 (1) ◽  
pp. 7-8 ◽  
Author(s):  
K. W. Chang
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Stability Result ◽  
The Real ◽  
Linear Differential ◽  
Image Position

Suppose that the real-valued function ƒ(t) is positive, continuous and monotonic increasing for t ≧ t0. If x = x (t) is a solution of the equation for for t ≧ t0, it is known that the solution x(t) oscillates infinitely often as t → ∞ and that the successive maxima of |x(t)| decrease, with increasing t. In particular x(t) is bounded as t → ∞.

Disconjugacy Conditions for the Third Order Linear Differential Equation

1969 ◽  
Vol 12 (5) ◽  
pp. 603-613 ◽  
Author(s):  
Lynn Erbe
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Sufficient Conditions ◽  
Order Linear Differential Equation ◽  
Third Order ◽  
Order Linear ◽  
The Third ◽  
Counting Multiplicity ◽  
Linear Differential ◽  
Image Position

An nth order homogeneous linear differential equation is said to be disconjugate on the interval I of the real line in case no non-trivial solution of the equation has more than n - 1 zeros (counting multiplicity) on I. It is the purpose of this paper to establish several necessary and sufficient conditions for disconjugacy of the third order linear differential equation(1.1)where pi(t) is continuous on the compact interval [a, b], i = 0, 1, 2.

On the Ordering of Multi-Point Boundary Value Functions

1970 ◽  
Vol 13 (4) ◽  
pp. 507-513 ◽  
Author(s):  
A. C. Peterson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Boundary Value ◽  
Value Functions ◽  
Order Linear Differential Equation ◽  
Point Boundary ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

We are concerned with the nth-order linear differential equation1where the coefficients are continuous. Aliev [1, 2] showed, in papers unavailable to the author that for n = 4(see Definition 2). Theorems 1 and 5 give respectively nth-order generalizations of these two results.

The asymptotic form of the Titchmarsh–Weyl m-function associated with a second order differential equation with locally integrable coefficient

1986 ◽  
Vol 102 (3-4) ◽  
pp. 243-251 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equationin the case where the only restriction on the real-valued function q is

An exact method for the calculation of certain Titchmarsh-Weyl m-functions

1987 ◽  
Vol 106 (1-2) ◽  
pp. 137-142 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Second Order ◽  
Exact Method ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe consider the second order, linear, differential equationwhere q is real-valued. In the case we calculate the Titchmarsh-Weyl m-function associated with (*).

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