Homogeneous Differential Equation Of Order Research Articles

SOLUTIONS OF ONE PROBLEM OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF THE SECOND ORDER

SOLUTIONS OF ONE PROBLEM OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF THE SECOND ORDER

SOLUTIONS OF ONE PROBLEM OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF THE SECOND ORDER

SOLUTIONS OF ONE PROBLEM OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF THE SECOND ORDER

A Method to Study the Cauchy Problem for a Singularly Perturbed Homogeneous Linear Differential Equation of Arbitrary Order

We construct a sequence converging to the solution to the Cauchy problem for a singularly perturbed linear homogeneous differential equation of any order. This sequence is asymptotic in the following sense: the distance (with respect to the norm of the space of continuous functions) between its nth element and the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter.

On a class of operators related to second-order differential equations

In the present paper, we find a class of linear homogeneous differential equations of order n + 1 (n > 1) whose fundamental system of solutions is constructed from the fundamental system of solutions of a second-order differential equation. The spectral properties of differential operators generated by these differential expressions are investigated. In particular, sufficient conditions are obtained for the coefficients of a second-order differential equation under which the case of maximal deficiency indices is realized.

Computing the Galois Group of a Linear Differential Equation of Order Four

In 1978 J. Kovacic described an efficient algorithm for computing liouvillian solutions of a linear homogeneous differential equation of order two over a field C(x), where x′ = 1 and C is an algebraically closed field of characteristic 0. During the years from 1990 to 1994 M. Singer and F. Ulmer published several papers in which they describe efficient algorithms for determining the Galois group of such a differential equation of order two or three and computing liouvillian solutions using this group. In this paper we present results concerning Galois groups of order four linear differential equations. In particular we construct a list of irreducible linear algebraic subgroups of SL(4, C) where C is an algebraically closed field of characteristic zero. This list is complete up to conjugation, and in the finite primitive case, up to projective equivalence. Then, in keeping with the spirit of the work of Kovacic, Singer and Ulmer we use representation theory to distinguish between the groups in this list.

Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions.

We show that the generating function for d-dimensional staircase polygons (by perimeter) can be expressed in terms of the generating function for the square of d-dimensional multinomial coefficients. This latter generating function is found to satisfy a linear, homogeneous differential equation of order d-1. This equation is solved for d\ensuremath{\le}4. For d=3 and d=4 the solution is obtained in terms of Heun functions, which are then shown to be expressible in terms of the complete elliptic integral of the first kind. The solutions are also shown to be related to lattice Green functions on three-dimensional lattices. The critical behavior of this model is determined exactly in all dimensions.

Geometry of matrix differential systems

In this paper we generalize some basic parts of the projective theory of linear homogeneous differential equations of order m, m > 2, from the scalar to the matrix case. In Section 1 we define the (m l)-dimensional left-projective space over the real n x n matrices P = P, ~ [(M,(R)) and, relying on a previous paper [7] and on standard methods of linear algebra, we state the basic properties of this space. In Section 2 the geometry of the matrix differential systems is described. To each basis of matrix solutions of a given system there corresponds a curve in P and a change of basis corresponds to a projective mapping of this curve. A given curve, together with its projective maps, corresponds to a class of differential systems having projectively equivalent solutions. Some simple properties of these curves are established and a new kind of disconjugacy is defined for the matrix differential systems.

Exact distributions of Hotelling's generalized T20

SUMMARY Percentile tables are given for the null distribution of Hotelling's generalized T2 statistic. These are obtained by analytic continuation of Constantine's series using a system of linear differential equations. The accuracy of certain approximations given by Pillai and Ito is discussed. T2= n2 tr (S1 S-1) = n2T, say, where S1 and S2 are independent m x m Wishart matrices on n1 and n2 degrees of freedom, respectively, estimating the same covariance matrix, with n2> >m. The exact null distribution when m = 2 was given by Hotelling in terms of the Gaussian hypergeometric function, and tabulated by F. E. Grubbs in 1954, in an unpublished report of Aberdeen Proving Ground, Maryland. In the general null case, the density function of T has been shown by the present author (1968) to satisfy a linear homogeneous differential equation of order m, with regular singularities at T = 0, -1, ..., - m and infinity. Constantine's (1966) series reduces in this case to the relevant solution of the differential equation in the unit circle about T = 0. Table 1 presents accurate percentiles of T2/n1 = (n2/n1) T, for m = 3 and 4 obtained by using an equivalent system of first order differential equations to carry out an analytic continuation of Constantine's series along the positive real axis. The accuracy of the procedure was checked by mapping the differential equation on to the unit interval using the change of variable Y = T/(T + 1). Agreement was generally to at least five significant figures. A similar tabulation has been carried out for m = 5.

The Relation Between the Zeros of a Solution of a Linear Homogeneous Differential Equation and those of its Derivatives

The fact that a non-identically vanishing solution of a linear homogeneous differential equation of order n and its first n 1 derivatives cannot all vanish at a point* raises the question as to the existence of a length so small that no solution and its first n 1 derivatives can all vanish within an interval of this length. In the first part of this paper I show that this question must be answered in the affirmative. This is done by determining such a length for any given equation. In the second part I consider in some detail such lengths in connection with equations of the second order. I also show that in certain cases there must be a root of the derivative of a solution of such an equation within a given distance from a root of the solution.