I am seeking a solution to an Abel equation of the first kind with $f_0=0$, $y’=f_3(x)y^3+f_2(x)y^2+f_1(x)y$

Question

The Abel equation of the first kind with $f_0=0$. $$ y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y \tag{1} $$ where \begin{align} f_3(x)&=(12x^2-2)/x,\\ f_2(x)&=(14x^2-1)/x^2,\\ f_1(x)&=3/x. \end{align} This equation originates from a boundary layer problem.

There exists solutions for specific Abel equations in Kamke's work [1], however, for constant and simple foefficients. Kamke [1] proposed the following procedure.

$$y(x)=E(x) G(z) \tag{2}$$

$$E(x)=\exp(\int f_1dx) \tag{3}$$ $$z= \int E(x)f_2dx \tag{4} $$ $$G'=g(z) G^3+G^2 \tag{5}$$ $$g(z)=\frac{E(x)f_3}{f_2} \tag{6}$$

Eq (2) is transformed as $$z(t)'=\frac{-1}{tG(z)} \tag{7}$$ $$t^2 z(t)''+g(z)=0 \tag{8}$$

Reference [2] purportedly provided a solution to Eq(8); however, I have identified inaccuracies in the presented solution, particularly in equations 4.13, 4.14, and 4.16. Despite my efforts to communicate with the authors to obtain a corrected version of the article, my email hasn't been successfully delivered. Moreover, equation 4.18 referenced in the text is not found in the cited reference [1, page 27) as indicated. I am currently unable to verify the impact of these errors on the final answer given in equation 4.23. This is why I am seek your help.

[1] E. Kamke, Differentialgleichungen, Losungsmethoden und Loesungen. I: Gewoehnliche Differentialgleichungen, Neunte Auflage, Mit einem Vorwort von Detlef Kamk, B. G. Teubner, Stuttgart, Germany, 1977.

[2] Dimitrios E. Panayotounakos, Theodoros I. Zarmpoutis, "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)", International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 387429, 13 pages, 2011. https://doi.org/10.1155/2011/387429

Answer 1

Judging by the discussion here https://math.stackexchange.com/questions/3761774/an-implementation-of-a-general-solution-of-abels-equations, you are not the only one who has had difficulties with the Panayotounakos - Zarmpoutis paper.

Note in particular the comment by E. S. Cheb-Terrab linked within the above discussion:

"...it is still the case that the authors of these papers do not show an ODE being solved with their method (their papers do not contain an example), and that the formula derived for the ODE (y'-1) y = f(x) does not verify even for the simplest forms of f(x)."

per (https://www.mapleprimes.com/posts/38740-Abel-ODEs-Were-Allegedly-Solved-In-2005).

FWIW, I attempted to solve your equation using Maxima, via the contrib_ode function. Among other methods, this function incorporates several methods for solving nonlinear equations, including Abel's equation. It could not find a solution, so I strongly suspect that your equation is not integrable.