On the characterization of exponential polynomials

SECTION 6 - Solutions of Linear 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.

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

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 probabilistic point of view on peak effects in linear difference equations

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 .

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].

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

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

A numerical method for finding the solution of linear time-varying differential equations is proposed. The properties of the hybrid functions, which consist of block-pulse functions plus rationalized Haar functions, are presented. The properties of the hybrid functions together with the operational matrices of integration and product are then utilized to reduce the solution of differential equations to the solution of algebraic equations. Examples are included to demonstrate the validity and applicability of the technique.

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

Investigating solutions of differential equations has been an important issue for scientists. Researchers around the world have talked about different methods to solve differential equations. The type and order of the differential equation enabled us to decide the method that we could choose to find the solution of the equation. One of these methods is the integral transform. Integral transform is the conversion of a real or complex valued function into another function by some algebraic operations. Integral transforms are used to solve many problems in mathematics and engineering, such as differential equations and integral equations. Therefore, new types of integral transforms have been defined and existing integral transforms have been improved. One of the solution methods of many physical problems also initial and boundary value problems are integral transforms. Integral transforms were introduced in the first half of the 19th century. The first historically known integral transforms are Laplace and Fourier transforms. Over the time, other transforms have emerged that are used in many fields. In this article, mohand transform was described and used to simplify the solution of linear ordinary differential equations with constant coefficients.

: Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.