Abel differential equation

2020 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)