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### Computing Divisors and Common Multiples of Quasi-linear Ordinary Differential Equations [chapter]

Dima Grigoriev, Fritz Schwarz
2013 Lecture Notes in Computer Science
We design an algorithm which finds first-order quasi-linear generalized divisors of a second-order quasi-linear ordinary differential equation.  ...  We prove that a quasi-linear common multiple of a pair of quasi-linear equations always exists and design an algorithm which yields a quasilinear common multiple.  ...  The first author is grateful to the Max-Planck Institut für Mathematik, Bonn for its hospitality during writing this paper and to Labex CEMPI (ANR-11-LABX-0007-01).  ...

### Methods in Mathematica for Solving Ordinary Differential Equations [article]

2011 arXiv   pre-print
An overview of the solution methods for ordinary differential equations in the Mathematica function DSolve is presented.  ...  We hope that the examples and ideas outlined in this paper will be useful for elementary and advanced courses on ordinary differential equations, as well as for solving differential equations which occur  ...  ) for solving differential equations.  ...

### On linearization of third-order ordinary differential equations

Sergey V Meleshko
2006 Journal of Physics A: Mathematical and General
A new algorithm for linearization of a third-order ordinary differential equation is presented.  ...  The algorithm consists of composition of two operations: reducing order of an ordinary differential equation and using the Lie linearization test for the obtained second-order ordinary differential equation  ...  New algorithm If the order of a third-order ordinary differential equation is reduced, then one can apply the Lie test for linearization.  ...

### Methods in Mathematica for Solving Ordinary Differential Equations

2011 Mathematical and Computational Applications
An overview of the solution methods for ordinary differential equations in the Mathematica function DSolve is presented.  ...  We hope that the examples and ideas outlined in this paper will be useful for elementary and advanced courses on ordinary differential equations, as well as for solving differential equations which occur  ...  ) for solving differential equations.  ...

### Page 400 of Mathematical Reviews Vol. , Issue 91A [page]

1991 Mathematical Reviews
ordinary differential equations.  ...  Some efficiently vectorizable algorithms for factorizing these types of matrices and solving the corresponding linear systems are described.  ...

### Symbolic Computation with Integro-Differential Operators

Georg Regensburger
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16
Differential operators already provide a rich algebraic structure with a wealth of results and algorithmic methods.  ...  In particular, we discuss normal forms, basic algebraic properties, and the computation of polynomial solutions for ordinary integro-differential equations with polynomial coefficients.  ...  For studying linear differential equations, the basic algebraic structure is the noncommutative ring of differential operators over a differential algebra.  ...

### Computing Highest-Order Divisors for a Class of Quasi-Linear Partial Differential Equations [chapter]

Dima Grigoriev, Fritz Schwarz
2015 Lecture Notes in Computer Science
We design an algorithm which for a class of quasi-linear partial differential polynomials of order k + 1 finds its quasi-linear divisors of order k.  ...  A differential polynomial G is called a divisor of a differential polynomial F if any solution of the differential equation G = 0 is a solution of the equation F = 0.  ...  The first author is grateful to the Max-Planck Institut für Mathematik, Bonn for its hospitality during writing this paper and to Labex CEMPI (ANR-11-LABX-0007-01).  ...

### Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one [article]

Jose Cano and Sebastian Falkensteiner and J. Rafael Sendra
2020 arXiv   pre-print
We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation.  ...  Moreover, if the reduced differential equation is non trivial, for every given point (x_0,y_0) in the complex plane, we prove the existence of a convergent Puiseux series solution y(x) of the original  ...  Note that in [4] it is shown that for general systems of algebraic ordinary differential equations the existence of non-constant formal power series solutions can not be decided algorithmically.  ...

### Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment

Irina Belyaeva, Igor Kirichenko, Oleh Ptashnyi, Natalia Chekanova, Tetiana Yarkho
2021 Eastern-European Journal of Enterprise Technologies
This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series  ...  The program for solving ordinary fourth-order differential equations could be used to construct Green's functions of boundary problems, to solve differential equations with private derivatives, a system  ...  Study [12] presented two algorithms for solving ordinary linear differential equations of the second order.  ...

### A Symbolic Framework for Operations on Linear Boundary Problems [chapter]

Markus Rosenkranz, Georg Regensburger, Loredana Tec, Bruno Buchberger
2009 Lecture Notes in Computer Science
For ordinary differential equations, the operations implemented include computing Green's operators, composing boundary problems and integrodifferential operators, and factoring boundary problems.  ...  We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system.  ...  We extended these results to a differential algebra setting in [4], where we also developed a factorization method applicable to boundary problems for ordinary differential equations (ODEs).  ...

### Page 4033 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews
ordinary differential equations by means of interval analytical methods.  ...  In this instructive paper the authors consider the initial value problem for a system of m ordinary differential equations of first order with an interval input vector and a perturbation parameter in the  ...

### Page 1243 of Mathematical Reviews Vol. , Issue 2001B [page]

2001 Mathematical Reviews
The two algorithms for obtaining the orthogonal matrix Q at step k (in factored form) are formu- 65L Ordinary differential equations 2001b:65075 lated and it is explained how to use them.  ...  The computation of the orthogonal factor Q of the QR factoriza- tion of some or all columns of the fundamental solution matrix Y of a linear system of ordinary differential equations dY/dt = A(t)Y is considered  ...

### Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

José Cano, Sebastian Falkensteiner, J. Rafael Sendra
2020 Mathematics in Computer Science
We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation.  ...  Moreover, if the reduced differential equation is non trivial, for every given point (x 0 , y 0 ) ∈ C 2 , we prove the existence of a convergent Puiseux series solution y(x) of the original system such  ...  To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.  ...

### Page 1145 of Mathematical Reviews Vol. , Issue 84c [page]

1984 Mathematical Reviews
Diestler, The discretization of continuous infinite sets of coupled ordinary linear differential equations: application to the collision-induced dissociation of a diatomic molecule by an atom (pp. 40-52  ...  Ortiz, Approximation of eigenvalues defined by ordinary differential equations with the tau method (pp. 90-102); Axel Ruhe, The two-sided Arnoldi algorithm for nonsymmetric eigenvalue problems (pp. 104  ...

### Computer Algebra Algorithms for Linear Ordinary Differential and Difference equations [chapter]

Manuel Bronstein
2001 European Congress of Mathematics
Galois theory has now produced algorithms for solving linear ordinary differential and difference equations in closed form.  ...  After introducing the relevant parts of the theory, we describe the latest algorithms for solving such equations.  ...  -A. Weil for their many useful comments and remarks during the preparation of this paper, as well as Prof. W. Decker for inviting me to present this survey at 3ecm.  ...
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