Riccati方程的可积条件

Integrable Conditions on Riccati Equation

黎卡提(Riccati)方程y′=p(x)y2+q(x)y+r(x)一般情况下是不可积的,本文利用变量变换先将Riccati方程化成二阶变系数齐线性方程,再化成二阶常系数线性方程.从而得到Riccati方程的若干可积条件.

Riccati equation y’=p(x)y_2+q(x)y+r(x) had no elementary solution in general case. In this paper, we first transfer the Riccati equation to second order linear homogenous equation with variable coefficients by variable transformation, and use the variable transformation to transfer the obtained second order linear homogeneous equation with variable coefficients to second order linear homogeneous equation with constant coefficients, then we obtain some integrable conditions on Riccati equation.