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**Constructing and solving equations – inverse operations.**
*(English)*
Zbl 1086.34006

This nice expository paper addresses an interesting problem regarding construction of equations (ordinary differential, partial differential, functional, etc.) having a given set of solutions along with the inverse problem of finding solutions for a given class of equations. Several important aspects of this problem are discussed, including, for instance, situations when the solution space has a finite number of parameters and solutions are not smooth enough, the solution space contains arbitrary functions as parameters and solutions are smooth, the solution space depends on arbitrary functions and solutions are not smooth enough, etc. A number of illuminating results and examples are presented and an open problem is stated. The paper concludes with final remarks and hints for further research.

Reviewer: Svitlana P. Rogovchenko (Famagusta)

### MSC:

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

34A30 | Linear ordinary differential equations and systems |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

39A12 | Discrete version of topics in analysis |

39B22 | Functional equations for real functions |

34A55 | Inverse problems involving ordinary differential equations |