Pairs of non-homogeneous linear differential polynomials

2006 ◽  
Vol 136 (4) ◽  
pp. 785-794 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Linear Differential Polynomials ◽  
Linear Differential ◽  
Image Position ◽  
Homogeneous Linear

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined by where k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

scholarly journals Note on the Peano-Baker Method of solving Linear Differential Equations

1915 ◽  
Vol 34 ◽  
pp. 41-44
Author(s):  
Arch Milne
Keyword(s):  
Differential Equations ◽  
Real Function ◽  
Linear Differential Equations ◽  
System Of Equations ◽  
The Matrix ◽  
Equivalent Equation ◽  
Linear Differential ◽  
Image Position ◽  
Homogeneous Linear

In Mathematische Annalen, Vol. 32 (1888) Peano discusses the solution of a system of homogeneous linear differential equationswhere rij denotes a real function of the variable t, and shows how, by a series of repeated substitutions, this system of equations may be replaced by the equivalent equationwhere X denotes the complex [x1, x2, … xn] and R the matrixof which equation the solution X can be represented as a sum of integrals.

Linear Differential Polynomials

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Linear Operators ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Linear Differential Operators ◽  
Differential Field ◽  
Linear Differential Polynomials ◽  
Linear Differential

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic results on them. In particular, it describes the property of a linear differential operator over a differential field K of defining a surjective map K → K, along with the transformation of a system of linear differential equations in several unknowns to an equivalent system of several linear differential equations in a single unknown. The chapter also discusses second-order linear differential operators, diagonalization of matrices, differential modules, linear differential operators in the presence of a valuation, and compositional conjugation. It concludes with an analysis of the Riccati transform and Johnson's Theorem.

The approach to scalar growth of a vector transformed by an increasing power of a matrix

1956 ◽  
Vol 52 (2) ◽  
pp. 213-214
Author(s):  
S. N. Afriat ◽  
H. G. Eggleston
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Image Position ◽  
Derivatives Of ◽  
Homogeneous Linear

The solution of a system of simultaneous homogeneous linear differential equations with constant coefficients has been obtained (1) in the formwhere the elements of the vector Z are the k functions determined by the system and their derivatives of order up to m – 1, where m is the order of the system. If Zn is the value of Z at t = n, then

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

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