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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.

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
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