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Substitution method for generalized linear differential equations

Dana Fraňková (1991)

Mathematica Bohemica

The generalized linear differential equation d x = d [ a ( t ) ] x + d f where A , f B V n l o c ( J ) and the matrices I - Δ - A ( t ) , I + Δ + A ( t ) are regular, can be transformed d y d s = B ( s ) y + g ( s ) using the notion of a logarithimc prolongation along an increasing function. This method enables to derive various results about generalized LDE from the well-known properties of ordinary LDE. As an example, the variational stability of the generalized LDE is investigated.

Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree.

Hector Giacomini, Malick Ndiaye (1996)

Publicacions Matemàtiques

In this paper, we consider polynomial systems of the form x' = y + P(x, y), y' = -x + Q(x, y), where P and Q are polynomials of degree n wihout linear part.For the case n = 3, we have found new sufficient conditions for a center at the origin, by proposing a first integral linear in certain coefficient of the system. The resulting first integral is in the general case of Darboux type.By induction, we have been able to generalize these results for polynomial systems of arbitrary degree.

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