The general form of Abel's equation of the first kind is given by:

Abel_ode := diff(y(x),x)=f3(x)*y(x)^3+f2(x)*y(x)^2+f1(x)*y(x)+f0(x);

$\mathrm{Abel\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

where f3(x), f2(x), f1(x) and f0(x) are arbitrary functions.

See Differentialgleichungen, by E. Kamke, p. 24. There is as yet no general solution for this ODE. For Abel's equation of the second kind, see Abel2A and Abel2C.

The most general method available at the moment to solve Abel ODEs seems to be the method of "Abel's invariant", described in E. Kamke, p. 26, as sub-method (g) due to M. Chini. The invariant of an Abel equation with f2=0 is the following quantity:

Abel_invariant := -1/27/f3(x)^4*(-diff(f0(x),x)*f3(x)+f0(x)*diff(f3(x),x)+
3*f0(x)*f3(x)*f1(x))^3/f0(x)^5;

$\mathrm{Abel\_invariant}≔-\frac{{\left(-\left(\frac{\ⅆ}{\ⅆx}\mathrm{f0}\left(x\right)\right)\mathrm{f3}\left(x\right)+\mathrm{f0}\left(x\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{f3}\left(x\right)\right)+3\mathrm{f0}\left(x\right)\mathrm{f3}\left(x\right)\mathrm{f1}\left(x\right)\right)}^3}{27{\mathrm{f3}\left(x\right)}^4{\mathrm{f0}\left(x\right)}^5}$

If the invariant does not depend on x, then the equation can be solved directly.

For an Abel equation with f2<>0, the f2 term can be removed by using the following transformation:

y(x)=u(x)-f2(x)/3/f3(x);

$y\left(x\right)=u\left(x\right)-\frac{\mathrm{f2}\left(x\right)}{3\mathrm{f3}\left(x\right)}$

The invariant can then be calculated as in the previous case. Note that if an Abel ODE has a constant invariant, then any other Abel ODE obtained from it by a transformation of the form

{y(x)=G(t)*u(t)+H(t), x=F(t)};

$\left\{x=F\left(t\right)\&comma;y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right)\right\}$

will also have a constant invariant (that is, is also solvable by this method).

The method "Chini" (see ?odeadvisor,Chini), also due to Chini, generalizes this method of the constant invariant for Abel ODEs.

$\mathrm{Abel\_ode}≔\mathrm{diff}\left(y\left(x\right)\&comma;x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

$\mathrm{Abel\_ode}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

$\mathrm{Abel\_invariant}≔-\frac{\frac{1}{27{\mathrm{f3}\left(x\right)}^4}{\left(-\mathrm{diff}\left(\mathrm{f0}\left(x\right)\&comma;x\right)\mathrm{f3}\left(x\right)+\mathrm{f0}\left(x\right)\mathrm{diff}\left(\mathrm{f3}\left(x\right)\&comma;x\right)+3\mathrm{f0}\left(x\right)\mathrm{f3}\left(x\right)\mathrm{f1}\left(x\right)\right)}^3}{{\mathrm{f0}\left(x\right)}^5}$

$\mathrm{Abel\_invariant}≔-\frac{{\left(-\left(\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{f0}\left(x\right)\right)\mathrm{f3}\left(x\right)+\mathrm{f0}\left(x\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{f3}\left(x\right)\right)+3\mathrm{f0}\left(x\right)\mathrm{f3}\left(x\right)\mathrm{f1}\left(x\right)\right)}^3}{27{\mathrm{f3}\left(x\right)}^4{\mathrm{f0}\left(x\right)}^5}$

$y\left(x\right)=u\left(x\right)-\frac{\mathrm{f2}\left(x\right)}{3\mathrm{f3}\left(x\right)}$

$y\left(x\right)=u\left(x\right)-\frac{\mathrm{f2}\left(x\right)}{3\mathrm{f3}\left(x\right)}$

$\left\{x=F\left(t\right)\&comma;y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right)\right\}$

$\left\{x=F\left(t\right)\&comma;y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right)\right\}$

$\mathrm{with}\left(\mathrm{DEtools}\&comma;\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{with}\left(\mathrm{PDEtools}\&comma;\mathrm{dchange}\right)$

$\left[\mathrm{dchange}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Abel\_ode}\right)$

$\left[\mathrm{\_Abel}\right]$

$\mathrm{ODE}≔\mathrm{diff}\left(y\left(x\right)\&comma;x\right)=\frac{\frac{1}{27}\left(12x+27x^3+27x^3{y\left(x\right)}^2+18x^{2}y\left(x\right)+27{y\left(x\right)}^3{x}^3+27x^2{y\left(x\right)}^2+9xy\left(x\right)+1\right)}{x^3}\&colon;$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ODE}\right)$

$\mathrm{ans}≔y\left(x\right)=\frac{29\mathrm{RootOf}\left(-81\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{841{\mathrm{\_a}}^3-27\mathrm{\_a}+27}\&DifferentialD;\mathrm{\_a}\right)+x+3\mathrm{c\_\_1}\right)x-3x-3}{9x}$

$\mathrm{TR\_LIN}≔\left\{x=F\left(t\right)\&comma;y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right)\right\}$

$\mathrm{TR\_LIN}≔\left\{x=F\left(t\right)\&comma;y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right)\right\}$

$\mathrm{ODE\_p}≔\mathrm{dchange}\left(\mathrm{TR\_LIN}\&comma;\mathrm{ODE}\&comma;\left[u\&comma;t\right]\right)\&colon;$

$\mathrm{ans\_p}≔\mathrm{dsolve}\left(\mathrm{ODE\_p}\&comma;u\left(t\right)\right)$

$\mathrm{ans\_p}≔u\left(t\right)=-\frac{9H\left(t\right)F\left(t\right)-29\mathrm{RootOf}\left(-81\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{841{\mathrm{\_a}}^3-27\mathrm{\_a}+27}\&DifferentialD;\mathrm{\_a}\right)+F\left(t\right)+3\mathrm{c\_\_1}\right)F\left(t\right)+3F\left(t\right)+3}{9G\left(t\right)F\left(t\right)}$

$\mathrm{ode}≔\mathrm{subs}\left(\left\{\mathrm{f0}\left(x\right)=0\&comma;\mathrm{f1}\left(x\right)=0\right\}\&comma;\mathrm{Abel\_ode}\right)$

$\mathrm{ode}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2$

$\mathrm{ITR}≔\left\{x=t\&comma;y\left(x\right)=\frac{\mathrm{f2}\left(t\right)}{\mathrm{f3}\left(t\right)}r\left(t\right)\right\}$

$\mathrm{ITR}≔\left\{x=t\&comma;y\left(x\right)=\frac{\mathrm{f2}\left(t\right)r\left(t\right)}{\mathrm{f3}\left(t\right)}\right\}$

$\mathrm{new\_ode}≔\mathrm{dchange}\left(\mathrm{ITR}\&comma;\mathrm{ode}\&comma;\left[r\left(t\right)\&comma;t\right]\right)\&colon;$

$\mathrm{constraint}≔\mathrm{diff}\left(\frac{\mathrm{f3}\left(t\right)}{\mathrm{f2}\left(t\right)}\&comma;t\right)-a\mathrm{f2}\left(t\right)=0$

$\mathrm{constraint}≔\frac{\frac{\&DifferentialD;}{\&DifferentialD;t}\mathrm{f3}\left(t\right)}{\mathrm{f2}\left(t\right)}-\frac{\mathrm{f3}\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}\mathrm{f2}\left(t\right)\right)}{{\mathrm{f2}\left(t\right)}^2}-a\mathrm{f2}\left(t\right)=0$

$\mathrm{new\_ode2}≔\mathrm{simplify}\left(\mathrm{new\_ode}\&comma;\left\{\mathrm{constraint}\right\}\&comma;\left\{\mathrm{diff}\left(\mathrm{f3}\left(t\right)\&comma;t\right)\right\}\right)$

$\mathrm{new\_ode2}≔\frac{-r\left(t\right){\mathrm{f2}\left(t\right)}^{3}a+\mathrm{f2}\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}r\left(t\right)\right)\mathrm{f3}\left(t\right)}{{\mathrm{f3}\left(t\right)}^2}=\frac{{\mathrm{f2}\left(t\right)}^3{r\left(t\right)}^2\left(r\left(t\right)+1\right)}{{\mathrm{f3}\left(t\right)}^2}$

$\mathrm{odeadvisor}\left(\mathrm{new\_ode2}\right)$

$\left[\mathrm{\_separable}\right]$

$\mathrm{ode}≔\mathrm{eval}\left(\mathrm{subs}\left(\left\{\mathrm{f0}\left(x\right)=4\&comma;\mathrm{f1}\left(x\right)=0\&comma;\mathrm{f2}\left(x\right)=\frac{1}{x}\&comma;\mathrm{f3}\left(x\right)=\frac{1}{x}\right\}\&comma;\mathrm{Abel\_ode}\right)\right)$

$\mathrm{ode}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=\frac{{y\left(x\right)}^3}{x}+\frac{{y\left(x\right)}^2}{x}+4$

$w\left(x\right)≔\exp \left(\mathrm{int}\left(\mathrm{f1}\left(x\right)-\frac{{\mathrm{f2}\left(x\right)}^2}{3\mathrm{f3}\left(x\right)}\&comma;x\right)\right)$

$w\left(x\right)≔{\&ExponentialE;}^{\int \left(\mathrm{f1}\left(x\right)-\frac{{\mathrm{f2}\left(x\right)}^2}{3\mathrm{f3}\left(x\right)}\right)\&DifferentialD;x}$

$\mathrm{tr}≔\left\{t=\mathrm{int}\left(\mathrm{f3}\left(x\right){'w'\left(x\right)}^2\&comma;x\right)\&comma;r\left(t\right)=\frac{\frac{1}{3\exp \left(\mathrm{int}\left(\mathrm{f1}\left(x\right)-\frac{\frac{1}{3}{\mathrm{f2}\left(x\right)}^2}{\mathrm{f3}\left(x\right)}\&comma;x\right)\right)}\left(3y\left(x\right)\mathrm{f3}\left(x\right)+\mathrm{f2}\left(x\right)\right)}{\mathrm{f3}\left(x\right)}\right\}$

$\mathrm{tr}≔\left\{t=\int \mathrm{f3}\left(x\right){w\left(x\right)}^2\&DifferentialD;x\&comma;r\left(t\right)=\frac{3y\left(x\right)\mathrm{f3}\left(x\right)+\mathrm{f2}\left(x\right)}{3{\&ExponentialE;}^{\int \left(\mathrm{f1}\left(x\right)-\frac{{\mathrm{f2}\left(x\right)}^2}{3\mathrm{f3}\left(x\right)}\right)\&DifferentialD;x}\mathrm{f3}\left(x\right)}\right\}$

$\mathrm{TR}≔\mathrm{eval}\left(\mathrm{subs}\left(\left\{\mathrm{f0}\left(x\right)=4\&comma;\mathrm{f1}\left(x\right)=0\&comma;\mathrm{f2}\left(x\right)=\frac{1}{x}\&comma;\mathrm{f3}\left(x\right)=\frac{1}{x}\right\}\&comma;\mathrm{tr}\right)\right)$

$\mathrm{TR}≔\left\{t=-\frac{3}{2x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\&comma;r\left(t\right)=\frac{x^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\frac{3y\left(x\right)}{x}+\frac{1}{x}\right)}{3}\right\}$

$\mathrm{ITR}≔\mathrm{solve}\left(\mathrm{TR}\&comma;\left\{x\&comma;y\left(x\right)\right\}\right)$

$\mathrm{ITR}≔\left\{x=-\frac{3\mathrm{RootOf}\left(2t{\mathrm{\_Z}}^2+3\right)}{2t}\&comma;y\left(x\right)=-\frac{\mathrm{RootOf}\left(2t{\mathrm{\_Z}}^2+3\right)-3r\left(t\right)}{3\mathrm{RootOf}\left(2t{\mathrm{\_Z}}^2+3\right)}\right\}$

$\mathrm{new\_ode}≔\mathrm{dchange}\left(\mathrm{ITR}\&comma;\mathrm{ode}\&comma;\left[t\&comma;r\left(t\right)\right]\&comma;'\mathrm{known}'=\mathrm{indets}\left(\mathrm{ode}\&comma;'\mathrm{unknown}'\right)\right)\&colon;$

$\mathrm{new\_ode2}≔\mathrm{simplify}\left(\mathrm{op}\left(1\&comma;\mathrm{map}\left(\mathrm{allvalues}\&comma;\left[\mathrm{solve}\left(\mathrm{new\_ode}\&comma;\left\{\mathrm{diff}\left(r\left(t\right)\&comma;t\right)\right\}\right)\right]\right)\right)\right)$

$\mathrm{new\_ode2}≔\left\{\frac{\&DifferentialD;}{\&DifferentialD;t}r\left(t\right)=\frac{18t^3{r\left(t\right)}^3-t^2\sqrt{6}\sqrt{-\frac{1}{t}}-243}{18t^3}\right\}$

$\mathrm{collect}\left(\mathrm{new\_ode2}\&comma;r\left(t\right)\&comma;\mathrm{factor}\right)$

$\left\{\frac{\&DifferentialD;}{\&DifferentialD;t}r\left(t\right)={r\left(t\right)}^3-\frac{t^2\sqrt{6}\sqrt{-\frac{1}{t}}+243}{18t^3}\right\}$