# 黎卡提方程

（重定向自Riccati方程

## 解法

${\displaystyle v'=v^{2}+P(x)v+Q(x)}$；其中令 ${\displaystyle Q(x)=q_{0}q_{2};P(x)=q_{1}+{\frac {q_{2}'}{q_{2}}}}$

1. ${\displaystyle v'=-\left({\frac {u'}{u}}\right)'=-{\frac {u''}{u}}+\left({\frac {u'}{u}}\right)^{2}=-{\frac {u''}{u}}+v^{2}\!}$

${\displaystyle {\frac {u''}{u}}=v^{2}-v'=-Q-Pv=-Q+P{\frac {u'}{u}}}$

${\displaystyle u''-Pu'+Qu=u''-(q_{1}+{\frac {q_{2}'}{q_{2}}})u'+q_{0}q_{2}u=0}$

## 施瓦茨方程上的應用

${\displaystyle S(w)\equiv \left({\frac {w''}{w'}}\right)'-{\frac {\left({\frac {w''}{w'}}\right)^{2}}{2}}=f}$

${\displaystyle y'-{\frac {y^{2}}{2}}=f}$

${\displaystyle u''-{\frac {1}{2}}fu=0}$

${\displaystyle w'={\frac {U'u-Uu'}{u^{2}}}}$
${\displaystyle w={\frac {U}{u}}}$

## 已知某一特定解

${\displaystyle z'+(q_{1}+2q_{2}y_{1})z=-q_{2}}$
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