An Abel ODE is a first order ODE of the form

 $y\'\left(x\right)=\mathrm{f0}\left(x\right)+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f3}\left(x\right){y\left(x\right)}^3$ 

The abelsol routine determines whether the first argument is a first order Abel ODE and, if so, attempts to find a closed form solution.

The first argument is a differential equation in diff or D form and the second argument is the function in the differential equation.

This function is part of the DEtools package, and so it can be used in the form abelsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[abelsol](..).

with(DEtools):

ode := diff(y(x),x)=a0+a1*y(x)+a2*y(x)^2+a3*y(x)^3;

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\mathrm{a0}+\mathrm{a1}y\left(x\right)+\mathrm{a2}{y\left(x\right)}^2+\mathrm{a3}{y\left(x\right)}^3$

abelsol( ode, y(x) );

ode := D(z)(t)+3*a*z(t)^3+6*a*t*z(t)^2;

$\mathrm{ode}≔\mathrm{D}\left(z\right)\left(t\right)+3a{z\left(t\right)}^3+6at{z\left(t\right)}^2$

abelsol( ode, z(t) );

$\left\{z\left(t\right)=\frac{1}{3a{t}^2+\mathrm{RootOf}\left(\mathrm{AiryBi}\left(\mathrm{\_Z}\right){\left(-3a\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{c\_\_1}t+{\left(-3a\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}t\mathrm{AiryAi}\left(\mathrm{\_Z}\right)+\mathrm{AiryBi}\left(1\,\mathrm{\_Z}\right)\mathrm{c\_\_1}+\mathrm{AiryAi}\left(1\,\mathrm{\_Z}\right)\right){\left(-3a\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right\}$