Global solutions of linear ordinary differential equations with polynomial coefficients

1975 ◽  
Vol 344 (1636) ◽  
pp. 51-64 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Power Series ◽  
Series Solution ◽  
Global Solutions ◽  
Linear Ordinary Differential Equation ◽  
Power Series Solution ◽  
Constructive Approach ◽  
Linear Ordinary Differential Equations ◽  
Polynomial Coefficients

A constructive approach is given, closely based on the work of Ford (1936) for continuing analytically a power series solution of a linear ordinary differential equation with polynomial coefficients outside the circle of convergence.

Langer's method for a second order linear ordinary differential equation with a double pole

1982 ◽  
Vol 91 (1) ◽  
pp. 111-118 ◽  
Author(s):  
Donatus Uzodinma Anyanwu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Series Solution ◽  
Asymptotic Series ◽  
Second Order ◽  
Linear Ordinary Differential Equation ◽  
Double Pole ◽  
Order Ordinary Differential Equation ◽  
Asymptotic Series Solution ◽  
Asymptotic Validity

If a second order ordinary differential equation has a simple or a double pole at a point zfl, then the standard Liouville-Green approximation could sometimes be valid near that point. In this paper we present an asymptotic series solution that is always valid near a double pole. A solution for that of a simple pole is also indicated. Asymptotic validity is proved.

On the differential invariants of a linear ordinary differential equation

1981 ◽  
Vol 89 (1-2) ◽  
pp. 111-123 ◽  
Author(s):  
Joseph P. S. Kung ◽  
Gian-Carlo Rota
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Invariant ◽  
Solution Space ◽  
Linear Ordinary Differential Equation ◽  
Differential Invariants ◽  
Linear Transformations ◽  
Differential Relation ◽  
Linear Ordinary Differential Equations ◽  
Algebraic Version

SynopsisLet y(n) + a1y(n−1 +…+ an−1y(1) + an = 0 (*) be a linear ordinary differential equation of order n. A (relative) differential invariant of (*) is a differential polynomial function π(xi) defined on the solution space of (*) satisfying: there is an integer g such that for all invertible linear transformations α of V into itself, π(αxi) = (det α)βπ(xi). We prove in a purely algebraic manner the following two theorems: A. The differential invariants of (*) are generated algebraically by the Wronskian W and the coefficients ala2, …, an of (*). B. Every generic differential relation (i.e. differential relation which holds for every linear ordinary differential equation of order n) among W, a1 …, an can be deduced algebraically from Abel's identity, W′ = −a1W. The second theorem may be considered as an algebraic version of the existence theorem for linear ordinary differential equations.

scholarly journals Using Homological Duality in Consecutive Pattern Avoidance

10.37236/2005 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Anton Khoroshkin ◽  
Boris Shapiro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Large Class ◽  
Generating Function ◽  
Generating Functions ◽  
Pattern Avoidance ◽  
Linear Ordinary Differential Equation ◽  
Sufficient Condition ◽  
Direct Algorithm ◽  
Polynomial Coefficients

Using an approach suggested by Dotsenko and Khoroshkin we present a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows us to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.

Analytic Solution to a Virus Infection Model

2013 ◽  
Vol 367 ◽  
pp. 503-507
Author(s):  
Liang Xu ◽  
Sha Xu
Keyword(s):  
Virus Infection ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Series Solution ◽  
Infection Model ◽  
Power Series Solution ◽  
Analytic Method ◽  
Linear Ordinary Differential Equations ◽  
Virus Infection Model

A functional analytic method was developed by E.K.Ifantis in 1987 to prove that certain non-linear ordinary differential equations (ODEs) have a unique power series solution which converges absolutely in a specified disc of the complex plane. In this paper, we first applied this method to certain systems of two non-linear ordinary differential equations. We proved that the power series solutions can be determined by some recurrence relations which depend on the parameters of the equations and the initial conditions. Then, we found a method to extend the range of the converge bound. At last, we applied the functional analytic method to the resistant virus infection model to obtain a power series solution and compared our solution with the numerical solution obtained by the Runge-Kutta method using the software Matlab (Version 7.0.1).

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