When working through the solution method found in Panayotounakos & Zarmpoutis's paper [1] on the Abel equation of the first kind I have come across a possible mistake which leaves the resulting general solution incorrect. 


I will match equation numbers as they appear in the paper where appropriate, it is linked below. For the Abel equation of the first kind in canonical form;
\begin{align}\tag{2.3a}\label{abel}
y_x'=y^3+f(x)^3,
\end{align}

three functions are introduced in order to write the general solution in terms of the Bessel functions. The subsidiary function $\Omega$:
\begin{align}\tag{2.6a}\label{Ric}
y'_x+fy^2&= \Omega,\\
y^3+fy^2+f^3&= \Omega,\tag{2.6b}
\end{align}
and the functions $f^*$ \& $g^*$:
\begin{align}\tag{2.9a}\label{ric}
f=-\frac{{f^*_x}'}{g^*},\\\tag{2.9b}
\Omega=-\frac{{g^*_x}'}{f^*},
\end{align}
so that \ref{Ric} may be written as a Riccati equation with known particular solution. In order to determine $f^*$ and $g^*$ Panayotounakos & Zarmpoutis eliminate $f^*$ in favor of $g^*$, then suppose the resulting ODE for $g^*$,
\begin{align}\tag{2.14a}\label{prob}
{g^*_{xx}}''-\frac{\Omega_x'}{\Omega}&{g^*_{x}}'-f\Omega g^*=0,\\
&f^*=\frac{{g^*_x}'}{\Omega},\tag{2.14b}
\end{align}

takes the form of the generalized Bessel ODE, 2.15. For this, it must hold that
\begin{align}\tag{2.17a}\label{sys}
\frac{G''_{xx}}{G'_x}-&\frac{G'_x}{G}=\frac{\Omega_x'}{\Omega},\\\label{sys2}
-\nu^2\left(\frac{G'_x}{G}\right)^2&+{G'_x}^2=-f\Omega,\tag{2.17b}
\end{align}
wherein lay the problem. The system of equations in \ref{sys} \& \ref{sys2} can be directly integrated as
\begin{align}\tag{*}\label{sol}
\Omega=C\frac{G'_x}{G}\implies\left(1-\frac{\nu^2}{G}\right)G'_x=-Cf\implies \frac12 G^2-\nu^2\log G=C'-C\int f\mathrm dx,
\end{align}
i.e. $G$ is entirely determined by $f$. And since $\Omega=y^3+fy^2+f^3$, this implies that
\begin{align}\tag{**}\label{cs}
y^3+fy^2+f^3=-C^2\frac{f}{G(\smallint f\mathrm dx)-\nu^2},
\end{align} 

which constitutes both a candidate solution to \ref{abel} and a constraint on the following results in [1]. I have plotted a few different functions for $f$ in \ref{cs} and compared to numerical solutions, they have not matched. Is this an error on my part or the papers? I have attempted to contact the email address provided with regards to my query, but it appears the email address is not working currently.



[This MO poster from early 2024][2] purports to find different errors later in the paper; equations 4.13, 4.14, and 4.16. Perhaps what I've identified above is the source? The accepted answer references posts with apparent troubles replicating solutions to Panayotounakos's 2005 paper [3], which I believe was erroneously equated to Panayotounakos and Zarmpoutis's 2011 paper [1]. The answer was helpful nonetheless. The veracity of the 2005 paper's results are a problem for a different post.


-----
[1] 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

[2] 

[3] Dimitrios E. Panayotounakos, "Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel’s equations), Applied Mathematics Letters, Volume 18, Issue 2, 2005, Pages 155-162, ISSN 0893-9659, https://doi.org/10.1016/j.aml.2004.09.004.


  [1]: https://doi.org/10.1155/2011/387429
  [2]: https://mathoverflow.net/questions/465908/i-am-seeking-a-solution-to-an-abel-equation-of-the-first-kind-with-f-0-0-y
  [3]: https://www.sciencedirect.com/science/article/pii/S0893965904003131?via%3Dihub