Difference between revisions of "Abel differential equation"
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− | + | {{MSC|34}} | |
+ | {{TEX|done}} | ||
− | + | 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 {{Cite|Ab}} on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural | ||
+ | generalization of the [[Riccati equation|Riccati equation]]. | ||
− | (Abel's differential equation of the first kind) | + | 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 $g_0(x) + g_1(x)y = 1/z$. |
− | + | 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, {{Cite|Go}}). | |
− | Abel's differential equations of the first and second kinds, as well as their further generalizations | ||
− | |||
− | |||
− | |||
− | have been studied in detail in the complex domain (see, for example, | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |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 | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Go}}||valign="top"| V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ka}}||valign="top"| E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint (1971) | ||
+ | |- | ||
+ | |} |
Latest revision as of 22:19, 4 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) |
Abel differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_differential_equation&oldid=27047