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Inverse problems in the theory of analytic planar vector fields. (English) Zbl 0926.34021

The authors state and analyze some new inverse problems in the theory of differential equations, i.e. the construction of an analytic planar vector field from a given finite number of solutions, trajectories or partial integrals. More precisely, the following problems are treated:
(1) the construction of an analytic vector field from a given number of solutions;
(2) the construction of an analytic vector field from a given complex analytic first integral;
(3) the construction of a vector field with given trajectories;
(4) the construction of a planar vector field from given algebraic partial integrals.
The last problem is illustrated by treating quadratic stationary vector fields with given algebraic curves of certain types.

MSC:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
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