SECTION 6 - Solutions of Linear Equations

Exponential polynomials, i.e. solutions of linear homogeneous differential equations with constant coefficients, are characterized as continuous solutions of linear homogeneous difference equations with constant coefficients. Namely, denote by C the \({\Bbb C}\)-linear space of all continuous functions on \({\Bbb R}\), and define for \( \omega \in {\Bbb R}_+\) the shift operator \(z_{\omega }:C\to C\) by \(\eta \mapsto z_{\omega }\eta \) such that \(z_{\omega }\eta (t)=\eta (t+{\omega })\) for all \(t\in {\Bbb R}\). Every ideal \({\mathfrak a}\) of \(R:={\Bbb C}[x,y]\) determines a subspace¶¶\( C_{\frak a}=\{\eta \in C:f(z_{{\omega }_1},z_{{\omega }_2})\eta =0\) for all \(f(x,y)\in {\frak a}\}\,.\)¶¶For incommensurable \({\omega }_1,{\omega }_2\in {\Bbb R}_+\) and ideals \({\frak a}\) in R with a factor ring \(R/{\frak a}\) , which is finite dimensional as a \({\Bbb C}\)-linear space, the set \(C_{\mathfrak a}\) is characterized. The main idea consists in applying the Kronecker approximation theorem to the analytic transition from the explicit representation of the solutions of recurrence sequences to that of the continuous solutions of difference equations. For an alternative proof, the Kronecker approximation theorem is combined with the Jordan decomposition theorem.

CHAPTER II - LINEAR DIFFERENTIAL EQUATIONS. SUPPLEMENTARY REMARKS ON THE THEORY OF DIFFERENTIAL EQUATIONS

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

In this paper, we derive new method for solving particular solution of linear second order ordinary differential equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct second solution of the associated homogeneous differential equation from this new method. Moreover, we have general solution method of linear second order ordinary differential equations without applying the two famous methods, undetermined coefficients method and variation of parameters method, for solving their particular solution

A definite theory exists for solving linear systems of ordinary differential equations with constant coefficients. “This theory is essentially a branch of linear algebra, and allows us to solve all autonomous linear equations”, ([4], p. 95). Furthermore, “In view of the fact that the solution of these equations does not, in principle, present any great difficulties, they are often considered to be of no great interest for theory, and in textbooks they are usually relegated to the position of simple exercises appended to the general theory of linear equations”, ([81], p. 41). Modern textbooks [8, 21, 26, 44, 45, 46, 50, 90] give a detailed and rigorous treatment of the standard methods available for linear problems, supplemented with numerous mechanical, electrical, and engineering applications. The voluminous literature accumulated on the general properties of the solutions of linear differential equations must therefore no doubt leave the impression that no further contribution could be made to the standard normal linear dynamic system with real and constant coefficients. This impression, however, will fade rather quickly as we go on. Paradoxically, some fundamental aspects of this simple dynamic system are still unnoticed.

A note on non-separable solutions of linear partial differential equations

In this paper the order and the hyper-order of the solutions of higher-order homogeneous linear differential equations is investigated.

The eigentensors of an arbitrary second-order tensor and their quality analyses

Solution of Homogeneous Linear Sequential Fractional Differential Equations with Constants Coefficients

In this paper, we study the growth of solutions of the second order linear complex differential equations insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

Journal of Mathematics and PhysicsVolume 29, Issue 1-4 p. 52-55 Article On the Explicit Numerical Solution of Linear Homogeneous Differential Equations with Constant Coefficients G. E. Hudson, G. E. Hudson Department of Physics, New York UniversitySearch for more papers by this author G. E. Hudson, G. E. Hudson Department of Physics, New York UniversitySearch for more papers by this author First published: April 1950 https://doi.org/10.1002/sapm195029152AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volume29, Issue1-4April 1950Pages 52-55 RelatedInformation

In this paper, we present conditions ensuring that solutions of linear second-order differential equations oscillate, provided solutions of corresponding difference equations oscillate. We also establish the converse result, namely, when oscillation of solutions of difference equations implies oscillation of solutions of corresponding differential equations.

Construction of a fundamental set of solutions of an arbitrary homogeneous linear difference equation

Generating solutions of a linear equation and structure of elements of the Zelisko group

The main aim of this chapter will be to use Theorem 4 of Chapter 2 – and specifically both existence and uniqueness of solutions – to develop a theory that will describe the solutions of the homogeneous linear second-order equation where p 0; p 1; p 2 are continuous real-valued functions on [a, b] and p 2(x) < 0 for each x in [a, b]. (‘Homogeneous’ here reflects the zero on the right-hand side of the equation which allows λy to be a solution (for any real constant λ) whenever y is a given solution.) The language of elementary linear algebra will be used and the theory of simultaneous linear equations will be presumed. Central to our discussion will be the Wronskian, or Wronskian determinant: if y 1 : [a, b] → ℝ and y 2 : [a, b] → ℝ are differentiable functions on the closed interval [a, b], the Wronskian of y 1 and y 2, W(y 1; y 2) : [a, b] → ℝ, is defined, for x ∈ [a, b], by If y 1 and y 2 are solutions of (1), it will turn out that either W(y 1, y 2) is identically zero or never zero in [a, b].