A Riccati ODE is a first order ODE of the form

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

The riccatisol routine determines whether the first argument is a first order Riccati ODE and, if so, returns a solution to the equation.

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

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

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{ode}≔x\mathrm{diff}\left(z\left(x\right)\,x\right)+z\left(x\right)=3x^2{z\left(x\right)}^2$

$\mathrm{ode}≔x\left(\frac{\ⅆ}{\ⅆx}z\left(x\right)\right)+z\left(x\right)=3x^2{z\left(x\right)}^2$

$\mathrm{riccatisol}\left(\mathrm{ode}\,z\left(x\right)\right)$

$\left\{z\left(x\right)=\frac{1}{\left(-3x+\mathrm{c\_\_1}\right)x}\right\}$

$\mathrm{ode}≔x\mathrm{D}\left(z\right)\left(x\right)+z\left(x\right)=3x^2{z\left(x\right)}^2-3\:$

$\mathrm{riccatisol}\left(\mathrm{ode}\,z\left(x\right)\right)$

$\left\{z\left(x\right)=\frac{-3+\frac{{\ⅇ}^{-6x}}{\mathrm{c\_\_1}+\frac{{\ⅇ}^{-6x}}{6}}}{3x}\right\}$