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CONVODE: a reduce package for solving differential equations

1993
*
Journal of Computational and Applied Mathematics
*

*The*package CONVODE (CONVersion

*of*Ordinary

*Differential*

*Equations*) is written in order to study

*differential*

*equations*and systems

*of*

*differential*

*equations*. ... ., CONVODE: a REDUCE package for solving

*differential*

*equations*, Journal

*of*Computational and Applied Mathematics 48 (1993) 157-165. ... It is evident that

*the*

*parameter*y1 associated to

*the*degree

*of*

*the*

*polynomial*, which is in

*the*

*differential*

*equation*, must have a numerically fixed entire value. ...

##
###
Page 7905 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

*The*author, in

*the*case d = 2, answers this question in

*the*negative by producing a family

*of*

*homogeneous*vector fields SLV;,

*depending*

*on*a

*parameter*/ € N, such that SLV, has no

*homogeneous*rational ... Summary: “In this paper, we discuss

*the*

*polynomial*

*solutions*

*of*algebraic

*differential*

*equations*. ...

##
###
Oscillation of Linear Ordinary Differential Equations: On a Theorem of A. Grigoriev

2006
*
Journal of dynamical and control systems
*

Grigoriev, placing an upper bound for

doi:10.1007/s10450-006-0008-8
fatcat:oi24crrokresharppp46kitv4m
*the*number*of*roots*of**linear*combinations*of**solutions*to systems*of**linear**equations*with*polynomial*or rational coefficients. ... We give a simplified proof and an improvement*of*a recent theorem by A. ... Any*linear*combination*of**solutions**of**the*system (4.1) satisfies a*linear*nth order*differential**equation**of**the*form (1.1) with*polynomial*coefficients a j ∈ C[t],*polynomially**depending**on**the**parameters*...##
###
Polynomial solutions of N th order non-homogeneous differential equations

2002
*
International Journal of Mathematical Education in Science and Technology
*

It was shown in 1 that

doi:10.1080/002073902321093577
fatcat:e5ielxwjrzbc5fd32tjwiwf3ba
*the**homogeneous**differential**equation*has a finite*polynomial**solution*if and only if where n is a root*of**the*recurrence relation. ... In this paper,*the*case in which*the**equation*has a forcing term*on**the*right hand side is considered. ... From*the*general theory*of**linear**differential**equations*, it is known that*the*general*solution**of*9 is: y Px i0 N c i y i x where y i x are n linearly independent*homogenous**solutions*. ...##
###
On Ordinary, Linear q-Difference Equations, with Applications to q-Sato Theory

2015
*
Journal of Operators
*

*The*purpose

*of*this paper is to develop

*the*theory

*of*ordinary,

*linear*q-difference

*equations*, in particular

*the*

*homogeneous*case; we show that there are many similarities to

*differential*

*equations*. ... In

*the*second part we study

*the*applications to a q-analogue

*of*Sato theory.

*The*q-Schur

*polynomials*act as basis function, similar to q-Appell

*polynomials*. ... We assume that since

*the*

*equation*is

*of*order , its general

*solution*will

*depend*

*on*distinct arbitrary constants and proceed to consider

*the*mode

*of*this

*dependence*. ...

##
###
Mellin–Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions

2012
*
Physics Letters B
*

We argue that

doi:10.1016/j.physletb.2012.06.045
fatcat:56kqwfgp3batpezhsm6eqokf74
*the*Mellin-Barnes representations*of*Feynman diagrams can be used for obtaining*linear*systems*of**homogeneous**differential**equations*for*the*original Feynman diagrams with arbitrary powers ... These systems*of**differential**equations*can be used (i) for*the**differential*reductions to sets*of*basic functions and (ii) for counting*the*numbers*of*master integrals. ... Any*polynomial*(rational)*solution**of*a multivariable*linear*system*of**differential**equations*related to a Feynman diagram can be written as a product*of**one*-loop bubble integrals and massless propagator ...##
###
Formal solutions of differential equations

1990
*
Journal of symbolic computation
*

P o w e r Series

doi:10.1016/s0747-7171(08)80037-5
fatcat:rz54tv4j3jezxjxugn22mr4kji
*Solutions**of**Differential*E q u a t i o n s My aim here is to contrast what is known*about**linear**differential**equations*with what is known*about*non-*linear**differential**equations*. ... Good general references for information*about**linear**differential**equations*are Poole (1960) and Schlesinger (1895). ... So far, we have only been considering*homogeneous**linear**differential**equations*, but*one*can ask*the*same questions*about*non-*homogeneous**linear**differential**equations*L(y) = b. ...##
###
Page 210 of Mathematical Reviews Vol. , Issue 90A
[page]

1990
*
Mathematical Reviews
*

*The*author gives theorems

*about*

*the*form

*of*particular

*solutions*

*of*fourth-order nonhomogeneous and

*homogeneous*ordinary differ- ential

*equations*

*of*Fuchs type, as an application

*of*

*the*fractional calculus ...

*The*constant N

*depends*

*on*A(t), r and é. L. J. ...

##
###
Page 75 of Mathematical Reviews Vol. 28, Issue 1
[page]

1964
*
Mathematical Reviews
*

*depending*

*on*z € Q and ¢ €[0, co). ... Under suitable restrictions

*on*

*the*functions C,(¢) and

*the*coefficients

*of*A, he proved that any

*solution*

*of*(1), sub- ject to certain

*homogeneous*boundary conditions, that 75 ...

##
###
Analytical solutions to second order differential equations

1977
*
Computers and Mathematics with Applications
*

A computer program, DIFFER, is described that will find analytical

doi:10.1016/0898-1221(77)90110-9
fatcat:3qs4uwqp7bcdnlgnrpwcnaiyfu
*solutions*to certain-second order ordinary*differential**equations*. ... An interactive main program is included which calls DIFFER and which will enable*the*user to run*the*program from a terminal. ...*Depending**on*which*of**the*4 expressions occur,*the*form*of**the*particular*solution*is chosen and substituted into*the**differential**equation*yielding a set*of*simultaneous*linear**equations*in which to solve ...##
###
Algorithms and methods in differential algebra

1996
*
Theoretical Computer Science
*

In this talk, I do not intend to give an exhaustive survey

doi:10.1016/0304-3975(95)00179-4
fatcat:qdnoqabglzbkxj4uz64udyff5y
*of*algorithmic aspects*of**Differential*Algebra but I only propose some ReSum4 L'algGbre diffirentielle a Ctt fond&e par J.F. ... Ritt,*Differential*Algebra is a true part*of*Algebra so that constructive and algorithmic problems and methods appear in this field. ... Acknowledgements It is a great pleasure for me to thank Jean-Marie Strelcyn for very helpful discussions*on*first integrals and especially for having drawn my attention to genericity results in this domain ...##
###
Integrability of Stochastic Birth-Death processes via Differential Galois Theory
[article]

2019
*
arXiv
*
pre-print

A system

arXiv:1901.06266v1
fatcat:wu5l7fazczcpbk2vwtjuo44v5y
*of*infinite, coupled ordinary*differential**equations*(*the*so-called master*equation*) describes*the*time-*dependence**of**the*probability*of*each system state. ... In this contribution we analyze*the*integrability*of*two types*of*stochastic birth-death processes (with*polynomial*birth and death rates) using standard*differential*Galois theory. ... Acknowledgements*The*authors kindly thank to*the*members*of*our Integrability Madrid Seminar for many fruitful discussions: Rafael Hernández-Heredero, Sonia Jiménez-Verdugo, Alvaro Pérez-Raposo, José Rojo-Montijano ...##
###
Multiparameter spectral theory

1968
*
Bulletin of the American Mathematical Society
*

In

doi:10.1090/s0002-9904-1968-11866-x
fatcat:jhnckmf4pzdofoqieis2edrec4
*one**of*these we allow nonlinear*dependence**on**the*scalar*parameter*, in particular,*polynomial*or rational*dependence*. In*the*matrix context, this is*the*topic*of*X-matrices (see e.g. [67] ). ... However,*the*exploitation*of*this area in*the*spirit*of*classical analysis, in particular for ordinary*differential**equations*, has reached no more than a preliminary stage. ... In*one**of*these we allow nonlinear*dependence**on**the*scalar*parameter*, in particular,*polynomial*or rational*dependence*. In*the*matrix context, this is*the*topic*of*X-matrices (see e.g. [67] ). ...##
###
Painlevé versus Fuchs

2006
*
Journal of Physics A: Mathematical and General
*

However, when there are certain restrictions

doi:10.1088/0305-4470/39/39/s16
fatcat:wpbecjserfaw5nvu7s4chjdm3i
*on**the*four*parameters*there exist*one**parameter*families*of**solutions*which do satisfy (Fuchsian)*differential**equations**of*finite order. ...*The*sigma form*of**the*Painlevé VI*equation*contains four arbitrary*parameters*and generically*the**solutions*can be said to be genuinely "nonlinear" because they do not satisfy*linear**differential**equations*...*One*can then deduce that*the**parameter*u is a*solution**of*a Riccati*differential**equation*. ...##
###
On Hamiltonian potentials with quartic polynomial normal variational equations
[article]

2012
*
arXiv
*
pre-print

In this paper we prove that there exists only

arXiv:0809.0135v2
fatcat:wz5vyo4nefgsfcl7flp6h25z2e
*one*family*of*classical Hamiltonian systems*of*two degrees*of*freedom with invariant plane Γ={q_2=p_2=0} whose normal variational*equation*around integral ... curves in Γ is generically a Hill-Schrödinger*equation*with quartic*polynomial*potential. ... Acknowledgements*The*research*of**the*first author is partially supported by grant FPI Spanish Government, project BFM2003-09504-C02-02. ...
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