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

1917 ◽  
Vol 18 (4) ◽  
pp. 214 ◽  
Author(s):  
William Benjamin Fite
Keyword(s):  
Differential Equation ◽  
Homogeneous Differential Equation ◽  
Linear Homogeneous Differential Equation

scholarly journals An integral equation associated with linear homogeneous differential equations

1986 ◽  
Vol 9 (2) ◽  
pp. 405-408 ◽  
Author(s):  
A. K. Bose
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Real Line ◽  
Homogeneous Differential Equation ◽  
The Real ◽  
Linear Homogeneous Differential Equation

Associated with each linear homogeneous differential equationy(n)=∑i=0n−1ai(x)y(i)of ordernon the real line, there is an equivalent integral equationf(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]duwhich is satisfied by each solutionf(x)of the differential equation.

scholarly journals ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

2020 ◽  
Vol 70 (2) ◽  
pp. 53-58
Author(s):  
P.B. Beisebay ◽  
◽  
G.H. Mukhamediev ◽  
Keyword(s):  
Differential Equation ◽  
Complex Analysis ◽  
Arbitrary Order ◽  
Fundamental System ◽  
Homogeneous Differential Equation ◽  
Complex Roots ◽  
Linear Homogeneous Differential Equation ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Homogeneous Linear

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

2019 ◽  
pp. 211-217 ◽  
Author(s):  
Vasiliy. I Fomin
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
General Solution ◽  
Operator Equation ◽  
Bounded Operator ◽  
Characteristic Operator ◽  
Homogeneous Differential Equation ◽  
Operator Functions ◽  
Linear Homogeneous Differential Equation ◽  
Real Argument

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied

About unbounded complex operators

2020 ◽  
pp. 57-67
Author(s):  
Vasiliy I. Fomin
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Explicit Form ◽  
Unbounded Operator ◽  
Inverse Operator ◽  
Complex Conjugate ◽  
Homogeneous Differential Equation ◽  
Linear Homogeneous Differential Equation

The concept of an unbounded complex operator as an operator acting in the pull-back of a Banach space is introduced. It is proved that each such operator is linear. Linear operations of addition and multiplication by a number and also the operation of multiplication are determined on the set of unbounded complex operators. The conditions for commutability of operators from this set are indicated. The product of complex conjugate operators and the properties of the conjugation operation are considered. Invertibility questions are studied: two contractions of an unbounded complex operator that have an inverse operator are proposed, and an explicit form of the inverse operator is found for one of these restrictions. It is noted that unbounded complex operators can find application in the study of a linear homogeneous differential equation with constant unbounded operator coefficients in a Banach space.

The number of real zeros of the solution of a linear homogeneous differential equation

1961 ◽  
Vol 57 (3) ◽  
pp. 693-694
Author(s):  
M. N. Brearley
Keyword(s):  
Differential Equation ◽  
Characteristic Equation ◽  
Homogeneous Differential Equation ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Linear Homogeneous Differential Equation ◽  
Constant Coefficients ◽  
Independent Variable ◽  
Image Position

The following theorem will be established:Provided the roots of the associated characteristic equation are all real, any solution of a linear homogeneous differential equation with constant coefficients has at most n − 1 zeros for real finite values of the independent variable, where n is the order of the equation.The theorem applies to equations with a complex independent variable, but since the conclusion concerns only real values of the variable there is no loss of generality in considering an equation of order n in the formwith y = y(x), where x is real. The constant coefficients ar may be complex.

scholarly journals On linear homogeneous differential equation of Chebyshev type

2008 ◽  
Vol 48 ◽  
Author(s):  
Eduard Kiriyatzkii
Keyword(s):  
Differential Equation ◽  
Fundamental System ◽  
Chebyshev System ◽  
Homogeneous Differential Equation ◽  
Linear Homogeneous Differential Equation

Let L[y] = y(n)(z)+gn-1(z)y(n-1)(z)+. . .+g1(z)y(1)(z)+g0(z)y(z) = 0  be a differential equation of nth order with analytic in circle |z| < R coefficients. We will call above equation by equation of Chebyshev type in |z| < R, if fundamental system of its solution is a Chebyshev system in circle |z| < R . In present paper the conditions with the fulfillment of which the equation L[y] = 0 is of Chebyshev type.  

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