The general form of Riccati's equation is given by the following:

Riccati_ode := diff(y(x),x)-f(x)*y(x)^2-g(x)*y(x)-h(x);

$\mathrm{Riccati\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)-f\left(x\right){y\left(x\right)}^2-g\left(x\right)y\left(x\right)-h\left(x\right)$

where $f\left(x\right)$, $g\left(x\right)$, and $h\left(x\right)$ are arbitrary functions.

The so-called "Special" form of Riccati's equation is given by:

Riccati_S := diff(y(x),x)+a*y(x)^2-b*x^c;

$\mathrm{Riccati\_S}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)+a{y\left(x\right)}^2-b{x}^c$

See Differentialgleichungen, by E. Kamke, pp. 21-23. There is as yet no general solution for the regular Riccati ODE.

For $h\left(x\right)=0$, we have an ODE of Bernoulli type which can be solved in a direct manner by dsolve (see odeadvisor,Bernoulli). In the other cases, a good strategy might be to introduce an appropriate change of variables. A general transformation for canceling $h\left(x\right)$ is not yet known.

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

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

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

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

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

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

$\mathrm{odeadvisor}\left(\mathrm{Riccati\_S}\right)$

$\left[\left[\mathrm{\_Riccati}\,\mathrm{\_special}\right]\right]$

$\mathrm{ITR}≔\left\{x=t\,y\left(x\right)=u\left(t\right)h\left(t\right)\right\}$

$\mathrm{ITR}≔\left\{x=t\,y\left(x\right)=u\left(t\right)h\left(t\right)\right\}$

$\mathrm{ode\_draft}≔\mathrm{dchange}\left(\mathrm{ITR}\,\mathrm{Riccati\_ode}\,\mathrm{known}=\left\{f\left(x\right)\,g\left(x\right)\,h\left(x\right)\right\}\,\left[t\,u\left(t\right)\right]\right)\:$

$\mathrm{ode2}≔\mathrm{diff}\left(u\left(t\right)\,t\right)=\mathrm{solve}\left(\mathrm{ode\_draft}\,\mathrm{diff}\left(u\left(t\right)\,t\right)\right)\:$

$\mathrm{new\_ode}≔\mathrm{collect}\left(\mathrm{ode2}\,u\left(t\right)\,\mathrm{normal}\right)$

$\mathrm{new\_ode}≔\frac{\ⅆ}{\ⅆt}u\left(t\right)=1+h\left(t\right)f\left(t\right){u\left(t\right)}^2+\frac{\left(h\left(t\right)g\left(t\right)-\frac{\ⅆ}{\ⅆt}h\left(t\right)\right)u\left(t\right)}{h\left(t\right)}$

$\mathrm{ITR}≔\left\{x=t\,y\left(x\right)=u\left(t\right)\exp \left(\mathrm{int}\left(g\left(t\right)\,t\right)\right)\right\}$

$\mathrm{ITR}≔\left\{x=t\,y\left(x\right)=u\left(t\right){\ⅇ}^{\int g\left(t\right)\ⅆt}\right\}$

$\mathrm{ode\_draft}≔\mathrm{dchange}\left(\mathrm{ITR}\,\mathrm{Riccati\_ode}\,\mathrm{known}=\left\{f\left(x\right)\,g\left(x\right)\,h\left(x\right)\right\}\,\left[t\,u\left(t\right)\right]\right)\:$

$\mathrm{ode2}≔\mathrm{isolate}\left(\mathrm{ode\_draft}\,\mathrm{diff}\left(u\left(t\right)\,t\right)\right)\:$

$\mathrm{new\_ode}≔\mathrm{collect}\left(\mathrm{ode2}\,u\left(t\right)\,\mathrm{simplify}\right)$

$\mathrm{new\_ode}≔\frac{\ⅆ}{\ⅆt}u\left(t\right)={\ⅇ}^{\int g\left(t\right)\ⅆt}f\left(t\right){u\left(t\right)}^2+h\left(t\right){\ⅇ}^{-\left(\int g\left(t\right)\ⅆt\right)}$

$\mathrm{dsolve}\left(\mathrm{subs}\left(c=0\,\mathrm{Riccati\_S}\right)\right)$

$y\left(x\right)=\frac{\tanh \left(\sqrt{ab}\left(\mathrm{c\_\_1}+x\right)\right)\sqrt{ab}}{a}$

$c\left[n\right]=\mathrm{seq}\left(-\frac{4i}{2i-1}\,i=-3..3\right)$

$c_n=\left(-\frac{12}{7}\,-\frac{8}{5}\,-\frac{4}{3}\,0\,\mathrm{-4}\,-\frac{8}{3}\,-\frac{12}{5}\right)$

$\mathrm{ode}≔\mathrm{subs}\left(c=-\frac{8}{3}\,\mathrm{Riccati\_S}\right)$

$\mathrm{ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)+a{y\left(x\right)}^2-\frac{b}{x^{\raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

$\mathrm{itr}≔\left\{x=r^{\frac{1}{c+3}}\,y\left(x\right)=\frac{1}{s\left(r\right)r^{\frac{2}{c+3}}}+\frac{1}{a{r}^{\frac{1}{c+3}}}\right\}$

$\mathrm{itr}≔\left\{x=r^{\frac{1}{c+3}}\,y\left(x\right)=\frac{1}{s\left(r\right)r^{\frac{2}{c+3}}}+\frac{1}{a{r}^{\frac{1}{c+3}}}\right\}$

$\mathrm{ITR}≔\mathrm{subs}\left(c=-\frac{8}{3}\,\mathrm{itr}\right)$

$\mathrm{ITR}≔\left\{x=r^3\,y\left(x\right)=\frac{1}{s\left(r\right)r^6}+\frac{1}{a{r}^3}\right\}$

$\mathrm{ode\_draft}≔\mathrm{dchange}\left(\mathrm{ITR}\,\mathrm{ode}\,\left[s\left(r\right)\,r\right]\right)\:$

$\mathrm{ode2}≔\mathrm{op}\left(\mathrm{solve}\left(\mathrm{ode\_draft}\,\left\{\mathrm{diff}\left(s\left(r\right)\,r\right)\right\}\right)\right)\:$

$\mathrm{new\_ode}≔\mathrm{expand}\left(\mathrm{combine}\left(\mathrm{ode2}\,\mathrm{symbolic}\right)\right)$

$\mathrm{new\_ode}≔\frac{\ⅆ}{\ⅆr}s\left(r\right)=-3b{s\left(r\right)}^2+\frac{3a}{r^4}$

$\mathrm{itr}≔\left\{x=r^{-\frac{1}{c+1}}\,y=\frac{1}{s\left(r\right)r^2+\frac{\left(c+1\right)r}{b}}\right\}$

$\mathrm{itr}≔\left\{x=r^{-\frac{1}{c+1}}\,y=\frac{1}{s\left(r\right)r^2+\frac{\left(c+1\right)r}{b}}\right\}$

$\mathrm{ode}≔\mathrm{subs}\left(h\left(x\right)=-f\left(x\right)-g\left(x\right)\,\mathrm{Riccati\_ode}\right)$

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

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

$y\left(x\right)=1-\frac{2{\ⅇ}^{\int \left(2f\left(x\right)+g\left(x\right)\right)\ⅆx}}{\mathrm{c\_\_1}+2\left(\int f\left(x\right){\ⅇ}^{\int \left(2f\left(x\right)+g\left(x\right)\right)\ⅆx}\ⅆx\right)}$

$\mathrm{convert}\left(\mathrm{Riccati\_ode}\,\mathrm{linearODE}\right)$

$\frac{{\ⅆ}^2}{\ⅆx^2}\mathrm{\_a}\left(x\right)=\frac{\left(g\left(x\right)f\left(x\right)+\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{\_a}\left(x\right)\right)}{f\left(x\right)}-f\left(x\right)h\left(x\right)\mathrm{\_a}\left(x\right),\left\{y\left(x\right)=-\frac{\frac{\ⅆ}{\ⅆx}\mathrm{\_a}\left(x\right)}{f\left(x\right)\mathrm{\_a}\left(x\right)}\right\}$