# Riccati equation

The nonlinear differential equation

$\frac{dy}{dx}}=f(x)+g(x)y+h(x){y}^{2$ | (1) |

is called Riccati equation. If $h(x)\equiv 0$, it is a question of a linear differential equation; if $f(x)\equiv 0$, of a Bernoulli equation. There is no general method for integrating explicitely the equation (1), but via the substitution

$$y:=-\frac{{w}^{\prime}(x)}{h(x)w(x)}$$ |

one can convert it to a homogeneous linear differential equation with non-constant coefficients.

If one can find a particular solution ${y}_{0}(x)$, then one can easily verify that the substitution

$y:={y}_{0}(x)+{\displaystyle \frac{1}{w(x)}}$ | (2) |

converts (1) to

$\frac{dw}{dx}}+[g(x)+2h(x){y}_{0}(x)]w+h(x)=\mathrm{\hspace{0.33em}0},$ | (3) |

which is a linear differential equation of first order with respect to the function $w=w(x)$.

Example. The Riccati equation

$\frac{dy}{x}}=\mathrm{\hspace{0.33em}3}+3{x}^{2}y-x{y}^{2$ | (4) |

has the particular solution $y:=3x$. Solve the equation.

We substitute $y:=3x+\frac{1}{w(x)}$ to (4), getting

$$\frac{dw}{dx}-3{x}^{2}w-x=\mathrm{\hspace{0.33em}0}.$$ |

For solving this first order equation (http://planetmath.org/LinearDifferentialEquationOfFirstOrder) we can put $w=uv$, ${w}^{\prime}=u{v}^{\prime}+{u}^{\prime}v$, writing the equation as

$u\cdot ({v}^{\prime}-3{x}^{3}v)+{u}^{\prime}v:=x,$ | (5) |

where we choose the value of the expression in parentheses equal to 0:

$$\frac{dv}{dx}-3{x}^{2}v=\mathrm{\hspace{0.33em}0}$$ |

After separation of variables^{} and integrating, we obtain from here a solution $v={e}^{{x}^{3}}$, which is set to the equation (5):

$$\frac{du}{dx}{e}^{{x}^{3}}=x$$ |

Separating the variables yields

$$du=\frac{x}{{e}^{{x}^{3}}}dx$$ |

and integrating:

$$u=C+\int x{e}^{-{x}^{3}}\mathit{d}x.$$ |

Thus we have

$$w=w(x)=uv={e}^{{x}^{3}}\left[C+\int x{e}^{-{x}^{3}}\mathit{d}x\right],$$ |

whence the general solution of the Riccati equation (4) is

$$y=\mathrm{\hspace{0.33em}3}x+\frac{{e}^{-{x}^{3}}}{C+\int x{e}^{-{x}^{3}}\mathit{d}x}.$$ |

It may be proved that if one knows three different solutions of Riccati equation (1), the each other solution may be expresses as a rational function of them.

Title | Riccati equation |
---|---|

Canonical name | RiccatiEquation |

Date of creation | 2013-03-22 18:05:43 |

Last modified on | 2013-03-22 18:05:43 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 12 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 34A34 |

Classification | msc 34A05 |

Synonym | Riccati differential equation |

Related topic | BernoulliEquation |