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### CONVODE: a reduce package for solving differential equations

A. Moussiaux
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

Sergei Yakovenko
2006 Journal of dynamical and control systems
Grigoriev, placing an upper bound for 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

Lawrence E. Levine, Ray Maleh
2002 International Journal of Mathematical Education in Science and Technology
It was shown in 1 that 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  Px   i0 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

Thomas Ernst
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

Mikhail Yu. Kalmykov, Bernd A. Kniehl
2012 Physics Letters B
We argue that 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

Michael. F Singer
1990 Journal of symbolic computation
P o w e r Series 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

S.J. Farlow
1977 Computers and Mathematics with Applications
A computer program, DIFFER, is described that will find analytical 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

Jean Moulin Ollagnier
1996 Theoretical Computer Science
In this talk, I do not intend to give an exhaustive survey 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]

Primitivo B. Acosta-Humanez, Jose A. Capitan, Juan J. Morales-Ruiz
2019 arXiv   pre-print
A system 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

F. V. Atkinson
1968 Bulletin of the American Mathematical Society
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] ).  ...  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

S Boukraa, S Hassani, J-M Maillard, B M McCoy, J-A Weil, N Zenine
2006 Journal of Physics A: Mathematical and General
However, when there are certain restrictions 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]

Primitivo B. Acosta-Humanez, David Blazquez-Sanz, Camilo Vargas Contreras
2012 arXiv   pre-print
In this paper we prove that there exists only 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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