Riccati型方程f'(y)dy/dx=P(x)f^2(y)+Q(x)f(y)+R(x)e^(∫Q(x)dx)的一个新的可积条件

A New Condition of Integrability of Equation of Riccati Type f'(y)dy/dx=P(x)f^2(y)+Q(x)f(y)+R(x)e^(∫Q(x)dx)

本文利用变量变换法与常数变易法给出Riccati型方程f'(y)dy/dx=P(x)f^2(y)+Q(x)f(y)+R(x)e^(∫Q(x)dx)的一个新的可积条件∫P(x)e^(∫Q(x)dx)dx=-1/2∫R(x)dx,同时给出该条件下方程的通解,并由此推得若干类Riccati方程的通解.

Using the change of variable and the variation of constant,this paper gives a new condition of integrability f＇（y）dy/dx=P（x）f^2（y）＋Q（x）f（y）＋R（x）e^∫Q（x）dx,for the equation of Riccati type f＇（y）dy/dx=P（x）f-2（y）＋Q（x）f（y）＋R（x）e^∫Q（x）dx,and the general solution of the equation under the condition.Based on the approach,general solutions of several kinds of Riccati equations are also formulated.