Riccati

Reduction to a second order linear equation

de http://en.wikipedia.org/wiki/Riccati_equation :

As explained on pages 23-25 of Ince's book, the non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE). Indeed if

$y'=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}$

then, wherever $q_{2}$ is non-zero, $v=yq_{2}$ satisfies a Riccati equation of the form

$v'=v^{2}+P(x)v+Q(x),$

where $Q=q_{2}q_{0}$ and $P=q_{1}+(q_{2}'/q_{2})$. In fact

$v'=(yq_{2})'=y'q_{2}+yq_{2}'=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+vq_{2}'/q_{2}=q_{0}q_{2}+(q_{1}+q_{2}'/q_{2})v+v^{2}.$

Substituting $v=-u'/u$, it follows that $u$ satisfies the linear 2nd order ODE

$u''-P(x)u'+Q(x)u=0$

since

$v'=-(u'/u)'=-(u''/u)+(u'/u)^{2}=-(u''/u)+v^{2}$

so that

$u''/u=v^{2}-v'=-Q-Pv=-Q+Pu'/u$

and hence

$u''-Pu'+Qu=0.$

A solution of this equation will lead to a solution $y=-u'/(q_{2}u)$ of the original Riccati equation.

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But then, the TDHO ODE is equivalent to the Riccati equation.

--Manuel-tepal 12:11 22 ago 2007 (CDT)