View source for Briot-Bouquet equation

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An ordinary differential equation

$$x^my'=f(x,y),\label{1}\tag{1}$$

where $m$ is a positive integer and the function $f$ is analytic at $x=y=0$, $f_y'(0,0)\neq0$, $f(0,0)=0$. It was shown by C. Briot and T. Bouquet [[#References|[1]]] that any equation of the type

$$\alpha(z,w)w'=\beta(z,w),$$

where $\alpha(0,0)=\beta(0,0)=0$ and $\alpha$ and $\beta$ 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 \eqref{1}. Equation \eqref{1} always (except for the case where $m=1$ and $f_y'(0,0)$ is a natural number) has a unique solution in the form of a formal power series:

$$y=\xi(x)\equiv\xi_1 x+\xi_2x^2+\dotsb,\label{2}\tag{2}$$

which converges for sufficiently small $x$ if $m=1$, and can diverge for all $x\neq0$ if $m>1$. In \eqref{1}, let

$$f\equiv f_0(x)+f_1(x)y,$$

then, for the series \eqref{2} to converge, it is necessary and sufficient to meet $m-1$ conditions concerning the coefficients of the Taylor series of $f_0$ and $f_1$; all the coefficients are included in these conditions, so that the existence or non-existence of an analytic solution $y=\xi(x)$ of equation \eqref{1} cannot be proved by any partial sum of the Taylor series of $f$ (cf. [[#References|[2]]], [[#References|[3]]]). For the case of a general function $f$ there are $(m-1)+(m-1)\times\infty$ such conditions, [[#References|[4]]]. Accordingly, the Briot–Bouquet equation is sometimes referred to as equation \eqref{1} with $m>1$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Briot,   T. Bouquet,   "Récherches sur les proprietés des équations différentielles"  ''J. École Polytechnique'' , '''21''' :  36  (1856)  pp. 133–198</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bieberbach,   "Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt" , Springer  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Martinet,   J.P. Ramis,   "Problèmes de modules pour des équations différentielles du premier ordre"  ''Publ. Math. IHES'' , '''55'''  (1982)  pp. 63–164</TD></TR></table>