Abel equation of the first kind

Example

Let $f_0, f_1, f_2, f_3$ be continuous functions on an open interval $I$, with $f_3$ not identically zero. The Abel equation of the first kind is the first-order ordinary differential equation $$y' = f_3(x)\, y^3 + f_2(x)\, y^2 + f_1(x)\, y + f_0(x).$$ Its right side is a cubic polynomial in $y$. When $f_3 = 0$ it would be a Riccati equation, so the Abel equation is the next case after the Riccati equation.

Instance: $y' = y^3$, with $f_3 = 1$ and $f_0 = f_1 = f_2 = 0$. It is separable, and its nonzero solutions are $y = \pm (C - 2x)^{-1/2}$ for $x < C/2$. Check: for $y = (C - 2x)^{-1/2}$, $y' = -\tfrac12 (C - 2x)^{-3/2} \cdot (-2) = (C - 2x)^{-3/2} = y^3$. The solution tends to infinity as $x \to C/2$, so it exists only on the interval $(-\infty, C/2)$, which is bounded on the right.