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 the set of zero coefficients of a function satisfying a linear differential equation

2012 ◽  
Vol 153 (2) ◽  
pp. 235-247 ◽  
Author(s):  
JASON P. BELL ◽  
STANLEY N. BURRIS ◽  
KAREN YEATS
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Recurrence ◽  
Nonzero Constant ◽  
Finite Set ◽  
Formula Group ◽  
Linear Differential ◽  
Image Position ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

AbstractLet K be a field of characteristic zero and suppose that f: → K satisfies a recurrence of the form \[ f(n) = \sum_{i=1}^d P_i(n) f(n-i), \] for n sufficiently large, where P1(z),. . .,Pd(z) are polynomials in K[z]. Given that Pd(z) is a nonzero constant polynomial, we show that the set of n ∈ for which f(n) = 0 is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem–Mahler–Lech theorem, which assumes that f(n) satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.

On the choice of standard solutions to Weber's equation

1952 ◽  
Vol 48 (3) ◽  
pp. 428-435 ◽  
Author(s):  
J. C. P. Miller
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Second Order ◽  
Linear Differential ◽  
Image Position ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation ◽  
Standard Solutions

AbstractIn this paper the principles for the choice of a pair of standard solutions of a homogeneous linear differential equation of the second order, described in an earlier paper (2), are applied to Weber's equation

Pinned-Pinned Slender Solid Struts with Parabolic Taper

1942 ◽  
Vol 46 (378) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Turton
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Elastic Stability ◽  
Formal Integration ◽  
Linear Differential ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

In 1917–19, Barling and Webb, Berry, Cowley and Levy, and Webb and Lang discussed the elastic stability of struts of various tapers, but it appears to have escaped notice that one of the few cases in which formal integration is possible is that in which the tapered profile of axial longitudinal sections is part of a parabola; this gives a “ homogeneous linear “ differential equation, i.e., a linear equation of the form f (xd/dx) y = F (x).

A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations

1950 ◽  
Vol 46 (4) ◽  
pp. 570-580 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Bessel Functions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Solutions ◽  
Linear Differential ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.

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