Determination of differential equations with associated solution and the possibility of building an active data base.

*(Italian. English summary)*Zbl 0732.34002Summary: The problem of building a differential equation jointly with its solutions is briefly analysed, and simple new methods are described for the particular case of linear homogeneous differential equations of the second order. This problem can be said to be marginal from a standard point of view, because one is usually interested in getting information concerning the solutions of a given equation. But the determination of equations together with their general solutions could become an interesting tool when considering the present development of computers and symbolic mathematical software. In fact one can imagine an active data base, where a large number of differential equations with their solutions are not only classified and registered, but also new equations and their integrals are generated, either automatically or in connection with some user requests. Starting with the above mentioned methods for building second-order equations, some features and problems of possible active bases for differential equations are considered and analysed.

##### MSC:

34A05 | Explicit solutions, first integrals of ordinary differential equations |

34A30 | Linear ordinary differential equations and systems, general |

65L99 | Numerical methods for ordinary differential equations |

68P15 | Database theory |

68W30 | Symbolic computation and algebraic computation |