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# Riccati equation

The nonlinear differential equation

$\displaystyle\frac{dy}{dx}\;=\;f(x)+g(x)y+h(x)y^{2}$ | (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 second order 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

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

converts (1) to

$\displaystyle\frac{dw}{dx}+[g(x)\!+\!2h(x)y_{0}(x)]\,w+h(x)\;=\;0,$ | (3) |

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

Example. The Riccati equation

$\displaystyle\frac{dy}{x}\;=\;3+3x^{2}y-xy^{2}$ | (4) |

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

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

$\frac{dw}{dx}-3x^{2}w-x\;=\;0.$ |

For solving this first order equation we can put $w=uv$, $w^{{\prime}}=uv^{{\prime}}+u^{{\prime}}v$, writing the equation as

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

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

$\frac{dv}{dx}-3x^{2}v\;=\;0$ |

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:

$u\;=\;C+\int xe^{{-x^{3}}}\,dx.$ |

Thus we have

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

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

$\displaystyle y\;=\;3x+\frac{e^{{-x^{3}}}}{C+\int xe^{{-x^{3}}}\,dx}.\\$ |

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.

## Mathematics Subject Classification

34A34*no label found*34A05

*no label found*

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