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

Abel_ode2C := (g1(x)*y(x)+g0(x))*diff(y(x),x)
= f3(x)*y(x)^3 + f2(x)*y(x)^2 + f1(x)*y(x) + f0(x);

$\mathrm{Abel\_ode2C}≔\left(\mathrm{g1}\left(x\right)y\left(x\right)+\mathrm{g0}\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^3+\mathrm{f2}\left(x\right){y\left(x\right)}^2+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$

where f3(x), f2(x), f1(x), f0(x), g1(x) and g0(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 28. There is as yet no general solution for this ODE.

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

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

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{dchange}\right)$

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

$\mathrm{ITR}≔\left\{x=t\,y\left(x\right)=\frac{1}{u\left(t\right)\mathrm{g1}\left(t\right)}-\frac{\mathrm{g0}\left(t\right)}{\mathrm{g1}\left(t\right)}\right\}$

$\mathrm{ITR}≔\left\{x=t\,y\left(x\right)=\frac{1}{u\left(t\right)\mathrm{g1}\left(t\right)}-\frac{\mathrm{g0}\left(t\right)}{\mathrm{g1}\left(t\right)}\right\}$

$\mathrm{new\_ode}≔\mathrm{dchange}\left(\mathrm{ITR}\,\mathrm{Abel\_ode2C}\,\left[u\left(t\right)\,t\right]\right)\:$

$\mathrm{new\_ode}≔\mathrm{collect}\left(\mathrm{diff}\left(u\left(t\right)\,t\right)=\mathrm{solve}\left(\mathrm{new\_ode}\,\mathrm{diff}\left(u\left(t\right)\,t\right)\right)\,u\left(t\right)\right)$

$\mathrm{new\_ode}≔\frac{\ⅆ}{\ⅆt}u\left(t\right)=\frac{\left(\mathrm{f3}\left(t\right){\mathrm{g0}\left(t\right)}^3-{\mathrm{g0}\left(t\right)}^2\mathrm{g1}\left(t\right)\mathrm{f2}\left(t\right)+\mathrm{g0}\left(t\right){\mathrm{g1}\left(t\right)}^2\mathrm{f1}\left(t\right)-\mathrm{f0}\left(t\right){\mathrm{g1}\left(t\right)}^3\right){u\left(t\right)}^3}{{\mathrm{g1}\left(t\right)}^2}+\frac{\left(-3\mathrm{f3}\left(t\right){\mathrm{g0}\left(t\right)}^2+2\mathrm{g0}\left(t\right)\mathrm{g1}\left(t\right)\mathrm{f2}\left(t\right)+\mathrm{g0}\left(t\right)\mathrm{g1}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{g1}\left(t\right)\right)-{\mathrm{g1}\left(t\right)}^2\mathrm{f1}\left(t\right)-{\mathrm{g1}\left(t\right)}^2\left(\frac{\ⅆ}{\ⅆt}\mathrm{g0}\left(t\right)\right)\right){u\left(t\right)}^2}{{\mathrm{g1}\left(t\right)}^2}+\frac{\left(3\mathrm{f3}\left(t\right)\mathrm{g0}\left(t\right)-\mathrm{g1}\left(t\right)\mathrm{f2}\left(t\right)-\mathrm{g1}\left(t\right)\left(\frac{\ⅆ}{\ⅆt}\mathrm{g1}\left(t\right)\right)\right)u\left(t\right)}{{\mathrm{g1}\left(t\right)}^2}-\frac{\mathrm{f3}\left(t\right)}{{\mathrm{g1}\left(t\right)}^2}$

$\mathrm{odeadvisor}\left(\mathrm{new\_ode}\,u\left(t\right)\,\left[\mathrm{Abel}\right]\right)$

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