The Abel equation of the first kind with f0=0. y'=f3(x)y^3+f2(x)y^2+f1(x)y Eq(1) where f3(x)=(12x^2-2)/x, f2(x)=(14x^2-1)/x^2, f1(x)=3/x. 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) Eq(2)

E(x)=exp(int f1dx) Eq(3) where (int=integral) z= int(E(x)f2dx) Eq(4) G'=g(z) G^3+G^2 Eq(5) g(z)=E(x)f3/f2 Eq(6) Eq (2) is transformed as z(t)'=-1/(tG(z)) Eq(7) t^2 z(t)''+g(z)=0 Eq(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 [![enter image description here][3]][3]

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