The general form of Abel's equation, second kind, class A is given by:

Abel_ode2A := (y(x)+g(x))*diff(y(x),x)=f2(x)*y(x)^2+f1(x)*y(x)+f0(x);

$\mathrm{Abel\_ode2A}≔\left(y\left(x\right)+g\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

where f2(x), f1(x), f0(x), and g(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 26. There is as yet no general solution for this ODE.

Note that all ODEs of type Abel, second kind, can be rewritten as ODEs of type Abel, first kind, as explained in ?odeadvisor,Abel2C

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{symgen}\,\mathrm{odeadvisor}\right)$

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

$\mathrm{odeadvisor}\left(\mathrm{Abel\_ode2A}\right)$

$\left[\left[\mathrm{\_Abel}\,\mathrm{2nd\; type}\,\mathrm{class\; A}\right]\right]$

$\mathrm{ode}≔\mathrm{eval}\left(\mathrm{subs}\left(\mathrm{f0}\left(x\right)=\mathrm{f1}\left(x\right)g\left(x\right)-\mathrm{f2}\left(x\right){g\left(x\right)}^2\,\mathrm{Abel\_ode2A}\right)\right)$

$\mathrm{ode}≔\left(y\left(x\right)+g\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+g\left(x\right)\mathrm{f1}\left(x\right)-\mathrm{f2}\left(x\right){g\left(x\right)}^2$

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\right)$

$y\left(x\right)=-g\left(x\right),y\left(x\right)=\left(\int -{\ⅇ}^{-\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\left(\mathrm{f2}\left(x\right)g\left(x\right)-\mathrm{f1}\left(x\right)\right)\ⅆx+\mathrm{c\_\_1}\right){\ⅇ}^{\int \mathrm{f2}\left(x\right)\ⅆx}$

$\mathrm{ode}≔\mathrm{eval}\left(\mathrm{subs}\left(\mathrm{f1}\left(x\right)=2\mathrm{f2}\left(x\right)g\left(x\right)-\mathrm{diff}\left(g\left(x\right)\,x\right)\,\mathrm{Abel\_ode2A}\right)\right)$

$\mathrm{ode}≔\left(y\left(x\right)+g\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=\mathrm{f2}\left(x\right){y\left(x\right)}^2+\left(2\mathrm{f2}\left(x\right)g\left(x\right)-\frac{\ⅆ}{\ⅆx}g\left(x\right)\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

$\mathrm{symgen}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=\frac{{\ⅇ}^{\int 2\mathrm{f2}\left(x\right)\ⅆx}}{y+g\left(x\right)}\right]$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,\mathrm{can}\right)$

$\mathrm{ans}≔y\left(x\right)=\frac{-{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}g\left(x\right)+\sqrt{{\left({\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\right)}^2{g\left(x\right)}^2+2{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\left(\int \frac{\mathrm{f0}\left(x\right)}{{\left({\ⅇ}^{\int \mathrm{f2}\left(x\right)\ⅆx}\right)}^2}\ⅆx\right)+2{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\mathrm{c\_\_1}}}{{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}},y\left(x\right)=-\frac{{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}g\left(x\right)+\sqrt{{\left({\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\right)}^2{g\left(x\right)}^2+2{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\left(\int \frac{\mathrm{f0}\left(x\right)}{{\left({\ⅇ}^{\int \mathrm{f2}\left(x\right)\ⅆx}\right)}^2}\ⅆx\right)+2{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}\mathrm{c\_\_1}}}{{\ⅇ}^{-2\left(\int \mathrm{f2}\left(x\right)\ⅆx\right)}}$