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The ordinary differential equation
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{{MSC|34}}
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{{TEX|done}}
  
<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/a/a010/a010110/a0101101.png" /></td> </tr></table>
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The ordinary differential equation
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$$
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y' = f_0(x) + f_1(x)y + f_2(x)y^2 + f_3(x)y^3
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$$
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(Abel's differential equation of the first kind) or
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$$
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\left(g_0(x) + g_1(x)y \right)y' = f_0(x) + f_1(x)y + f_2(x)y^2 + f_3(x)y^3
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$$
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(Abel's differential equation of the second kind). These equations arose in the context of the studies of N.H. Abel {{Cite|Ab}} on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural
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generalization of the [[Riccati equation|Riccati equation]].
  
(Abel's differential equation of the first kind) or
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If $f_1 \in C(a,b)$ and $f_2,f_3 \in C^1(a,b)$ and $f_3(x) \neq 0$ for $x \in [a,b]$, then Abel's differential equation of the first kind can be reduced to the normal form $\mathrm{d}z/\mathrm{d}t = z^3 + \Phi(t)$ by substitution of variables {{Cite|Ka}}. In the general case, Abel's differential equation of the first kind cannot be integrated in closed form, though this is possible in special cases {{Cite|Ka}}. If $g_0,g_1 \in C^1(a,b)$ and $g_1(x) \neq 0$, $g_0(x) + g_1(x)y \neq 0$, Abel's differential equation of the second kind can be reduced to Abel's differential equation of the first kind by substituting $g_0(x) + g_1(x)y = 1/z$.
  
<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/a/a010/a010110/a0101102.png" /></td> </tr></table>
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Abel's differential equations of the first and second kinds, as well as their further generalizations  
 
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$$
(Abel's differential equation of the second kind). These equations arose in the context of the studies of N.H. Abel [[#References|[1]]] on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural generalization of the [[Riccati equation|Riccati equation]].
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y' = \sum_{i=0}^n f_i(x)y^i, \quad
 
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y' \sum_{j=0}^m g_j(x)y^j = \sum_{i=0}^n f_i(x)y^i,
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101105.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101106.png" />, then Abel's differential equation of the first kind can be reduced to the normal form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101107.png" /> by substitution of variables [[#References|[2]]]. In the general case, Abel's differential equation of the first kind cannot be integrated in closed form, though this is possible in special cases [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a0101109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a01011010.png" />, Abel's differential equation of the second kind can be reduced to Abel's differential equation of the first kind by substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010110/a01011011.png" />.
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$$
 
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have been studied in detail in the complex domain (see, for example, {{Cite|Go}}).
Abel's differential equations of the first and second kinds, as well as their further generalizations
 
 
 
<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/a/a010/a010110/a01011012.png" /></td> </tr></table>
 
 
 
have been studied in detail in the complex domain (see, for example, [[#References|[3]]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"N.H. Abel,   "Précis d'une théorie des fonctions elliptiques" ''J. Reine Angew. Math.'' , '''4''' (1829) pp. 309–348</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Golubev,  "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR></table>
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{|
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|valign="top"|{{Ref|Ab}}||valign="top"| N.H. Abel, "Précis d'une théorie des fonctions elliptiques" ''J. Reine Angew. Math.'', '''4''' (1829) pp. 309–348
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|-
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|valign="top"|{{Ref|Go}}||valign="top"| V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
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|valign="top"|{{Ref|Ka}}||valign="top"| E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint (1971)
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|-
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|}

Latest revision as of 00:19, 5 July 2012

2010 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

The ordinary differential equation $$ y' = f_0(x) + f_1(x)y + f_2(x)y^2 + f_3(x)y^3 $$ (Abel's differential equation of the first kind) or $$ \left(g_0(x) + g_1(x)y \right)y' = f_0(x) + f_1(x)y + f_2(x)y^2 + f_3(x)y^3 $$ (Abel's differential equation of the second kind). These equations arose in the context of the studies of N.H. Abel [Ab] on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural generalization of the Riccati equation.

If $f_1 \in C(a,b)$ and $f_2,f_3 \in C^1(a,b)$ and $f_3(x) \neq 0$ for $x \in [a,b]$, then Abel's differential equation of the first kind can be reduced to the normal form $\mathrm{d}z/\mathrm{d}t = z^3 + \Phi(t)$ by substitution of variables [Ka]. In the general case, Abel's differential equation of the first kind cannot be integrated in closed form, though this is possible in special cases [Ka]. If $g_0,g_1 \in C^1(a,b)$ and $g_1(x) \neq 0$, $g_0(x) + g_1(x)y \neq 0$, Abel's differential equation of the second kind can be reduced to Abel's differential equation of the first kind by substituting $g_0(x) + g_1(x)y = 1/z$.

Abel's differential equations of the first and second kinds, as well as their further generalizations $$ y' = \sum_{i=0}^n f_i(x)y^i, \quad y' \sum_{j=0}^m g_j(x)y^j = \sum_{i=0}^n f_i(x)y^i, $$ have been studied in detail in the complex domain (see, for example, [Go]).

References

[Ab] N.H. Abel, "Précis d'une théorie des fonctions elliptiques" J. Reine Angew. Math., 4 (1829) pp. 309–348
[Go] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[Ka] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1971)
How to Cite This Entry:
Abel differential equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Abel_differential_equation&oldid=27047
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article