From some point of view, after Riccati type equations, the simplest first order ordinary differential equations (ODEs) are those having as right hand side (RHS) a third degree polynomial in the dependent variable, also called Abel type ODEs:

PDEtools[declare](y(x), f(x), prime=x);  # turn ON the enhanced DE display

diff(y(x),x) = f[3](x)*y(x)^3 + f[2](x)*y(x)^2 + f[1](x)*y(x) + f[0](x);

where { f[1], f[2] f[3] } are arbitrary functions of x. Abel equations appear in the reduction of order process related to finding exact solutions to many second and higher order ODE families, and hence are frequently found in the modelling of real problems in varied areas. That has for a long time motivated people to study their integrable cases and related solving methods.

The implementation of methods for Abel ODEs dsolve follows the presentation done in Abel Equations: Equivalence and New Integrable Classes, by E.S. Cheb-Terrab and A.D. Roche, Computer Physics Communications 130 (2000).

A general "exact integration" strategy for these ODEs was first formulated by Liouville, in then 19th century, and is based on the concepts of classes, invariants and the solving of the equivalence problem. Generally speaking, two Abel ODEs belong to the same equivalence class if and only if one can be obtained from the other by means of a transformation of the form

TR := {x = F(t), y(x)= P(t)*u(t) + Q(t)};

where t and u(t) are respectively the new independent and dependent variables, and F, P, Q are arbitrary functions of t satisfying

diff(F(x),x)<>0, P <> 0;

Each class has infinitely many member and there are infinitely many classes. To each class there corresponds a different set of values of the invariants (built with the coefficients {f[1], f[2], f[3] } and their derivatives), and actually any one of them (we shall pick one and call it the invariant) is enough to characterize a class.

A simple integrable case happens when the invariant is constant; the solution to the ODE then follows straightforwardly in terms of quadratures, as explained in textbooks and in odeadvisor, Abel (the method for constant invariant Abel ODEs was implemented in Maple Release 5).

On the contrary, when the invariant is not constant, just a few integrable cases are known and the formulation of solving strategies is entirely based on the equivalence between two such Abel ODEs (one of which is integrable) under the transformation TR.

When a given Abel ODE belongs to one of a set of the solvable classes collected in the aforementioned CPC paper, dsolve's routines first determine this fact, without solving any differential equations, and use it to return a closed form solution without requiring further participation from the user. Apart from new solvable Abel classes, the ODE families that are covered include, as particular cases, all the Abel solvable cases presented in Kamke's and Murphy's books.

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The relative invariant s3 is: 10/27/x^12*(9*x^2+2)
The first absolute invariant s5^3/s3^5 is: -729/100*(90*x^4-15*x^2-14)^3/(9*x^2+2)^5
The second absolute invariant s3*s7/s5^2 is: 5/3*(9*x^2+2)*(972*x^6-324*x^4-15*x^2+98)/(90*x^4-15*x^2-14)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
inverse of the transformation solving the problem is: {t = x, u(t) = y(x)}
<- Abel successful

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The first invariant is non-rational: -729/100*(90*F(x)^4-15*F(x)^2-14)^3/(9*F(x)^2+2)^5
-> Searching for a convenient change of variables...
<- Unable to rationalize the invariant
The relative invariant s3 is: 10/27*(9*F(x)^2+2)/F(x)^12*diff(F(x),x)^3*P(x)^3
The first absolute invariant s5^3/s3^5 is: -729/100*(90*F(x)^4-15*F(x)^2-14)^3/(9*F(x)^2+2)^5
The second absolute invariant s3*s7/s5^2 is: 5/3*(9*F(x)^2+2)*(972*F(x)^6-324*F(x)^4-15*F(x)^2+98)/(90*F(x)^4-15*F(x)^2-14)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
inverse of the transformation solving the problem is: {t = F(x), u(t) = P(x)*y(x)+Q(x)}
<- Abel successful

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The equivalent Abel ODE of 1st kind is: diff(u(x),x) = -2*x*(x^2+1)*(x^2-1)*u(x)^3-2/x^2*u(x)^2-1/x*u(x)
The relative invariant s3 is: -8/27*(27*x^8-9*x^4+2)/x^6
The first absolute invariant s5^3/s3^5 is: 729*(135*x^16-36*x^12+54*x^8-15*x^4+2)^3/(27*x^8-9*x^4+2)^5
The second absolute invariant s3*s7/s5^2 is: 1/3*(27*x^8-9*x^4+2)*(2835*x^24+243*x^20+2349*x^16-927*x^12+495*x^8-105*x^4+10)/(135*x^16-36*x^12+54*x^8-15*x^4+2)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
...checking Abel class AIA (by Halphen)
...checking Abel class AIL (205)
...checking Abel class AIA (147)
...checking Abel class AIL (581)
...checking Abel class AIL (200)
...checking Abel class AIL (257)
inverse of the transformation solving the problem is: {t = 1/x^2, u(t) = x*y(x)}
<- Abel successful