# Sur la méthode de l’intégrale particulière et sur ses conséquences pour l’équation de Riccati et pour les équations différentielles linéaires et homogènes d’ordre supérieur

Commentationes Mathematicae (2007)

- Volume: 47, Issue: 2
- ISSN: 2080-1211

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topAndrzej Kapcia. "Sur la méthode de l’intégrale particulière et sur ses conséquences pour l’équation de Riccati et pour les équations différentielles linéaires et homogènes d’ordre supérieur." Commentationes Mathematicae 47.2 (2007): null. <http://eudml.org/doc/291543>.

@article{AndrzejKapcia2007,

abstract = {This paper presents the method of particular solution for solving the Riccati equation and linear homogenous equations of second and third order, as well as its certain application to linear homogenous equations of n-th order. The conditions of effective integrability for equations (0.1) and (0.2) are expressed in symbolic (operator) form and also for equation (0.3) in fully expanded form. There have been proved three theorems which state the following: for any subclass of differential equations of the form (0.1), (0.2), (0.3), if there are known, respectively: a particular solution $y_0$, a particular solution u 0 , two linearly independent particular solutions $u_1 , u_2$, then it is possible to construct superclasses of differential equations of the given class, using classes cited in [6, 7, 8, 9]. Moreover, one may obtain their effectively integrable generalizations. Numerous examples provided illustrate the above results. The article presents also a practical way of applying the method of particular solution to linear equations of n-th order. This method enables us to integrate more general equations than those described in [4, 5, 14] of the form (0.1), (0.2), (0.3), (0.4) for which the particular solutions are cited therein.},

author = {Andrzej Kapcia},

journal = {Commentationes Mathematicae},

keywords = {differential equation; linear; homogenous; order; particular solution; superclass; subclass; inverse operator; linearly independent; effective integrability; general solution},

language = {eng},

number = {2},

pages = {null},

title = {Sur la méthode de l’intégrale particulière et sur ses conséquences pour l’équation de Riccati et pour les équations différentielles linéaires et homogènes d’ordre supérieur},

url = {http://eudml.org/doc/291543},

volume = {47},

year = {2007},

}

TY - JOUR

AU - Andrzej Kapcia

TI - Sur la méthode de l’intégrale particulière et sur ses conséquences pour l’équation de Riccati et pour les équations différentielles linéaires et homogènes d’ordre supérieur

JO - Commentationes Mathematicae

PY - 2007

VL - 47

IS - 2

SP - null

AB - This paper presents the method of particular solution for solving the Riccati equation and linear homogenous equations of second and third order, as well as its certain application to linear homogenous equations of n-th order. The conditions of effective integrability for equations (0.1) and (0.2) are expressed in symbolic (operator) form and also for equation (0.3) in fully expanded form. There have been proved three theorems which state the following: for any subclass of differential equations of the form (0.1), (0.2), (0.3), if there are known, respectively: a particular solution $y_0$, a particular solution u 0 , two linearly independent particular solutions $u_1 , u_2$, then it is possible to construct superclasses of differential equations of the given class, using classes cited in [6, 7, 8, 9]. Moreover, one may obtain their effectively integrable generalizations. Numerous examples provided illustrate the above results. The article presents also a practical way of applying the method of particular solution to linear equations of n-th order. This method enables us to integrate more general equations than those described in [4, 5, 14] of the form (0.1), (0.2), (0.3), (0.4) for which the particular solutions are cited therein.

LA - eng

KW - differential equation; linear; homogenous; order; particular solution; superclass; subclass; inverse operator; linearly independent; effective integrability; general solution

UR - http://eudml.org/doc/291543

ER -

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