Briot-Bouquet equation

An ordinary differential equation

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

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 [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 \ref{1}. Equation \ref{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:

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

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

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

then, for the series \ref{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 \ref{1} cannot be proved by any partial sum of the Taylor series of $f$ (cf. [2], [3]). For the case of a general function $f$ there are $(m-1)+(m-1)\times\infty$ such conditions, [4]. Accordingly, the Briot–Bouquet equation is sometimes referred to as equation \ref{1} with $m>1$.

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