Nth Order Linear Differential Equations Research Articles

Asymptotic Behavior of Solutions to Some nTh Order Linear Differential Equations

where gj(t), j=O, , n-2, are continuous on [to, oo), in terms of the known functions zi (t), * * *, Zn (t). Specifically, we seek conditions in order that for each solution u(t) of (2), there exist constants Aj, j = 1, * * *, n, such that (3) u (t) = Al(t)z (t) + + An(t)z (t) i = 0, ** *,- 1, and

Lyapunov function having the stability matrix of Hermite in its time derivative

A method developed to generate Lyapunov functionals for partial differential equations is applied to an nth-order ordinary linear differential equation. The resulting function V has the Hermite stability matrix in its derivative V. The function is efficient as the sufficient stability conditions obtained from V are also necessary.

On solutions of 𝑁th order linear differential equations with 𝑁 zeros

On solutions of 𝑁th order linear differential equations with 𝑁 zeros

Onn th order equations having critical degreen − 2

In this paper a fundamental set of solutions is determined for certain nth order linear differential equations.

Extended Reducibility of Some Differential Operators

where G(D, x) is a linear polynomial operator of order less than that of P(D, x); F will then have to be a polynomial of degree ? 2. In'a previous paper [1], the following operational identities were established: (1) x D( _ [(xD + 1 n)(xD + 1 2n) ... (xD + 1 rn)D]n, (2) x(r?l)nDrn= [x(xD + 1 n) (xD + 1 2n) ... (xD + 1 rn)]n, (3) D(r+l)nXrn = [(xD + 1 + n)(xD + 1 + 2n) ... (xD + 1 + rn)D]n, (4) Drnx(?+l)n=[x (xD + 1 + n)(xD + 1 + 2n) ... (xD + 1 + rn)]n. The above equations imply the reducibility of the operators given on the lefthand side. In addition, by means of the reducibility, certain nth order linear differential equations were solved. In this paper, we extend the notion of reducibility. We then exhibit a class of operators satisfying this condition and apply it to solving certain other nth order linear differential equations. First, let us consider a special case of (1), i.e., (5) xnD2n {xD2 + (1 n)D}In. The idea for extending the notion of reducibility was obtained from attempts to ascribe a meaning to (5) for nonintegral n. If, in (5), we let n -> n + 2, we formally obtain ( 6) X~~n+2-D2n+1 _{D2 + ( n)}(2n+l),2 (6) x = {xD 2 -In ( D We will show that (6) holds if it is interpreted as (7) = {xD2 (2 -)D To establish (7), we let x = eZ. Since XDx= DZ(DZ -1) ** (D, n + 1), we have to show that

An existence theorem for linear stochastic differential equations

The purpose of this paper is to prove a general existence and uniqueness theorem (Theorem 1) for nth order linear stochastic differential equations in which derivatives are defined in the “mean-square” sense (see below for details). The results here subsume the earlier work of Edwards and Moyal [I], and the present proofs are simpler. Further, it is hoped that the present treatment exhibits the structure (and limitations) of linear stochastic differential equations in the mean-square or strong sense. This is discussed and contrasted with the “pointwise” or It6 equations in the last section. Actually, it will be seen from the present paper that the theory of linear stochastic differential equations in the strong sense merges with the theory of ordinary linear differential equations in Banach spaces, the latter spaces reducing to the scalars only when the underlying probability space contains just one point. Precise definitions of the initial value problem and of the relevant concepts will be given in the next section. The two-point boundary value problem is treated in Section 3, and a result on the sample function behavior of solutions is obtained in Section 4.

Generalized Rodriques formula solutions for certain linear differential equations

It was found that, when this equation has a positive integer root n, (1) has solutions which can be expressed in terms of a generalized Rodrigues formula having n 1 for the index of differentiation. For nonpositive integer roots of (2) a general solution for (1) can be given by an iterated indefinite integral and for nonintegral roots the corresponding solution is a contour integral which reduces to the Rodrigues formula for integer roots. The contour integral result was obtained only for the homogeneous form of (1). The purpose of this paper is to present the remaining results obtained by the author in [3], namely the extension of the above results for secondorder equations to certain nth-order linear differential equations. The equation to be discussed is

On the Equivalence of Discrete Systems in Time-Optimal Control

It is shown in this paper that under a transformation which maps the state space of a system onto a sequence space in a one-to-one manner, time-optimal discrete regulator systems with different dynamics and modulators can be brought to equivalence and that this fact may be used to construct an optimal control strategy for one system from the known optimal control strategy of another one. In particular, the optimal control strategy for a pulse-width-modulated discrete system with a plant described by an nth order linear differential equation is derived in this manner.

Characterization of time-varying linear systems

Abstract : It is shown that a versatile description of time-varying linear systems is achieved by the use of a family of eight system functions that includes the G and H transfer functions. This approach is extended to multidimensional systems and to ''vector-time'' systems operating on signals that are specified by a function of several variables. The class of time-varying linear systems that is specified implicitly by a set of nth order linear differential equations in r variables is considered. The explicit form of the impulse response of the system is derived in order to bridge the gap between the differential equation and the system-function specifications of a time-varying linear system. The same basic approach for characterizing deterministic systems leads to procedures for the characterization of randomly varying systems subjected to random inputs. The general case where the system and input are nonstationary is considered for both unidimensional and multidimensional systems. (Author)

Solution of Systems of Linear Ordinary Differential Equations with Periodic Coefficients

An analysis technique is presented to provide an essentially explicit solution for a system of n simultaneous first-order linear differential equations with periodic coefficients. This representation of a periodic variable-parameter linear system of arbitrary finite order is chosen for its theoretical and practical advantages over the classical nth order linear differential equation. Emphasis is placed on natural mode solutions of a homogeneous set of equations. The characteristic exponents for these solutions are determined from a polynomial equation the coefficients of which are linear combinations of n — 1 convergent infinite-order determinants. Approximate calculation of these determinants is feasible for problems of moderate order.

A Note on the Indical Equation

INTRODUCTION A useful technique for solving linear differential equations is the Frobenius series expansion in which it is assumed that the form of the solution is k=oo (1) y = (x-a)C ' dk(x-.a)k k=o The coefficients are determined by substituting (1) into the differential equation and settingthe coefficients of each power of (x-a) equal to zero. The exponent c of the prefactor is found from what is called the indical equation which is usually obtained by considering the lowest power of (x-a). The object of this note is to present a method for finding c based on L'Hospital's rule [1] for evaluating an indeterminate expression. While the general nth order linear differential equation

A Note on an nth Order Linear Differential Equation

(1) (x-a)n(x-b)nDny = Ay, (a4 b), has been solved previously by Halphen (see Kamke, Differentialgleichungen, p. 541) by means of the substitution y = (x-b)n-)xF(loga) which transforms Eq. (1) into a linear one in F with constant coefficients. We effect the solution here in a much simpler way. Also, we find the limit of the solution as b -+ a. We assume a solution of the form y = (x-a)m(x-6b)n-m,. On substituting back in Eq. (1) and applying Liebniz's rule for the differentiation of a product, we find that m must be any root (mr) of the nth order polynomial equation

Generalized immittance kernels and the Kronig-Kramers relations

A general relation between the poles and residues of a Kronig-Kramers pair is established which enables one to prove that the generalized immittance kernel obtained as the solution of an Nth-order linear differential equation with constant coefficients and sinusoidal forcing must necessarily be consistent with the Kronig-Kramers relations. Finally, it is proved that the network or system function of a system exhibiting a distribution over any or all of the (N + 1) arbitrary parameters of the Nth-order immittance kernel must itself be consistent with the Kronig-Kramers relations. The example of Lorentz dispersion is discussed.

On the relation between Green’s functions and covariances of certain stochastic processes and its application to unbiased linear prediction

As a result of the stimulus of the work of Wiener [1](2) and Kolmogoroff [2] the theory of linear prediction has developed during the past decade. In this paper we shall develop a method of solution for a class of problems which are natural generalizations of these that have been treated by the socalled auto-correlation theory as developed by Phillips and Weiss [3] or Cunningham and Hynd [4]. A basic difficulty in a linear prediction theory employing continuous observations appears to be that of solving the integral equations of the first kind that arise from the minimization of the error variance. Since, as in the auto-correlation theory, the prediction is based on a finite past history, such tools as Wiener's generalized Fourier analysis are not available and a general method of solution of the integral equations has not yet been devised. Here we develop a method of solution that requires hypotheses which are satisfied in many important physical applications. Moreover, as will be seen from Part I of the present paper, the kernel of our method is based on the intrinsically interesting relationship that exists between covariance functions of random processes generated by driving nth order linear differential equations by so-called pure noise and the Green's function of a suitably defined self-adjoint equation of order 2n. For physically stable linear differential equations with constant coefficients, it will be shown (Corollary 1.1) that such covariance functions are in fact Green's functions of a suitably defined self-adjoint problem. For linear differential equations with variable coefficients this is no longer true but such covariance