几类特殊的黎卡提(Riccati)方程通解求法

General Solutions of Some Special Riccati Equations

能用初等方法求解的微分方程是很有限的 ,例如形式上很简单的黎卡提 (Riccati)方程dydx =P(x) y2 +Q(x) y +R(x) ( 1 )对一般的函数P ,Q ,R而言 ,其通解不可能用初等函数或初等函数的积分予以表示。当然对一般的非线性方程更是如此。本文仅就P(x) ,Q(x) ,R(x)存在一定内在联系时 ,黎卡提方程 ( 1 )及黎卡提方程 ( 2 )dydx =P(x) y3+Q(x) y2 +R(x) y+m(x) ( 2 )

In this paper the problem of general solutions for the Riccati equations d y d x=P(X)y 2+Q(x)y+R(x) and d y d x=P(X)y 3+Q(x)y 2+R(x)y+M(x) is studied. Here P(x),Q(x),R(x),M(x) have some internal relations.