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Difference between revisions of "Briot-Bouquet equation"

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An ordinary differential equation
 
An ordinary differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$x^my'=f(x,y),\tag{1}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176502.png" /> is a positive integer and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176503.png" /> is analytic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176506.png" />. It was shown by C. Briot and T. Bouquet [[#References|[1]]] that any equation of the type
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176507.png" /></td> </tr></table>
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$$\alpha(z,w)w'=\beta(z,w),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b0176509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765012.png" /> is a natural number) has a unique solution in the form of a formal power series:
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$y=\xi(x)\equiv\xi_1 x+\xi_2x^2+\dots,\tag{2}$$
  
which converges for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765015.png" />, and can diverge for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765017.png" />. In (1), let
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which converges for sufficiently small $x$ if $m=1$, and can diverge for all $x\neq0$ if $m>1$. In \ref{1}, let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765018.png" /></td> </tr></table>
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$$f\equiv f_0(x)+f_1(x)y,$$
  
then, for the series (2) to converge, it is necessary and sufficient to meet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765019.png" /> conditions concerning the coefficients of the Taylor series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765021.png" />; all the coefficients are included in these conditions, so that the existence or non-existence of an analytic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765022.png" /> of equation (1) cannot be proved by any partial sum of the Taylor series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765023.png" /> (cf. [[#References|[2]]], [[#References|[3]]]). For the case of a general function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765024.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765025.png" /> such conditions, [[#References|[4]]]. Accordingly, the Briot–Bouquet equation is sometimes referred to as equation (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017650/b01765026.png" />.
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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. [[#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 \ref{1} with $m>1$.
  
 
====References====
 
====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>
 
<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>

Revision as of 15:23, 30 August 2014

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:

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

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
How to Cite This Entry:
Briot-Bouquet equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Briot-Bouquet_equation&oldid=22189
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article