Inverse problems in the theory of analytic planar vector fields (Q1283340)

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.