Briot-Bouquet equation
An ordinary differential equation
![]() | (1) |
where is a positive integer and the function
is analytic at
,
,
. It was shown by C. Briot and T. Bouquet [1] that any equation of the type
![]() |
where and
and
are analytic at the origin, can be reduced, by means of a special local changes of the variables, to a finite number of equations of type (1). Equation (1) always (except for the case where
and
is a natural number) has a unique solution in the form of a formal power series:
![]() | (2) |
which converges for sufficiently small if
, and can diverge for all
if
. In (1), let
![]() |
then, for the series (2) to converge, it is necessary and sufficient to meet conditions concerning the coefficients of the Taylor series of
and
; all the coefficients are included in these conditions, so that the existence or non-existence of an analytic solution
of equation (1) cannot be proved by any partial sum of the Taylor series of
(cf. [2], [3]). For the case of a general function
there are
such conditions, [4]. Accordingly, the Briot–Bouquet equation is sometimes referred to as equation (1) with
.
References
[1] | C. Briot, T. Bouquet, "Récherches sur les proprietés des équations différentielles" J. École Polytechnique , 21 : 36 (1856) pp. 133–198 |
[2] | L. Bieberbach, "Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt" , Springer (1965) |
[3] | A.D. Bryuno, "Analytical form of differential equations. Introduction" Trans. Moscow Math. Soc. , 25 (1971) pp. 134–151 Trudy Moskov. Mat. Obshch. , 25 (1971) pp. 120–138 |
[4] | J. Martinet, J.P. Ramis, "Problèmes de modules pour des équations différentielles du premier ordre" Publ. Math. IHES , 55 (1982) pp. 63–164 |
Briot-Bouquet equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Briot-Bouquet_equation&oldid=22189