Abel differential equation

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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   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  . 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  . 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,  ).

References

	N.H. Abel, "Précis d'une théorie des fonctions elliptiques" J. Reine Angew. Math., 4 (1829) pp. 309–348
	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1971)