An implementation of a general solution of Abel's equations

Question

Introduction

An Abel equation of the second kind in its canonical form is writen as

$$y(x)y'(x)-y(x)=f(x)\quad (6)$$ for arbitrary $f(x)$. This equation has a general solution derived by 1 2 (Check Wikipedia article for Abel equation of the first kind, which in some cases reduces to the above form)

What I am trying to do is to solve the equation

$$y(x)y'(x)-y(x)=Ax^2 $$ over the reals with $A$ constant. This equation does not appear in the tabulated solutions of Polyanin, Manzhirov, so I tried to implement the method from the paper 1. However, I noticed that the solution does not agree with the numerics, so I tried to run my code on an equation that we know the solution to, which is:

$$y(x)y'(x)-y(x)=x $$

The result is summarised below using the language of Mathematica. As we can see, the analytical plot does not belong to the solution family seen in the second picture.

Goal

I am trying to write a code that would apply the method described in 1 to solve the general Abel equation. The solution can be applied in any programming language.

Method

Code

(*Solving for y*)
ζ[x_] = Log[Abs[x + 2 λ]]
F[x_] = ζ[x]
G[ζ_] = 
1/16 ((ζ Sin[ζ] + Cos[ζ]) CosIntegral[ζ] + 
 Cos[ζ]^2)*(4 ζ CosIntegral[ζ] + 
  Cos[ζ] )/(ζ CosIntegral[ζ])^3 Exp[-ζ] - 
2 F[ζ]

a = -4;
b[ζ_] = 3 + 4 (G[ζ] + F[ζ]) Exp[-ζ]
c[ζ_] = - 4 (G[ζ] + 2 F[ζ]) Exp[-ζ]

p[ζ_] = -a^2/3 + b[ζ]
q[ζ_] = 2 (a/3)^3 - a b[ζ]/3 + c[ζ]
NN = Solve[Z^3 + p Z + q == 0, Z] /. {p -> p[ζ], 
 q -> q[ζ]} /. ζ -> ζ[x];
y = Table[
Table[1/2 (x + 2 λ) ((Z /. NN[[ii]]) + 1/3), {ii, 1, 
 3}], {λ, 1, 2}];

(*Benchmarking*)
DSolve[Y'[X] Y[X] - Y[X] == X, Y[X], X]

p1 = Plot[y, {x, 0, 10}]
p2 = Table[
ContourPlot[
1/10 ((5 + Sqrt[5]) Log[
      1 + Sqrt[5] - (2 Y)/X] - (-5 + Sqrt[5]) Log[-1 + Sqrt[5] + (
       2 Y)/X]) == II - Log[X], {X, 0, 10}, {Y, -10, 15}], {II, 0,
 3, 0.2}];
Show[p2]

Plots

The plots look not alike

References

1. Dimitrios E. Panayotounakos, "Exact analytic solutions of unsolvable classes of first and second" order nonlinear ODEs (Part I: Abel’s equations)
2. Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I, "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)".