一类Riccati微分方程的解法

Solution of a Class of Riccati Differential Equations

一般形式的Riccati微分方程,虽然形式简单,但却不存在初等解法.又因其应用广泛,因此对该方程的具体求解仍具有研究意义.根据方程系数函数的特点,由把方程转化为变量分离方程这一思想,确定了方程中系数函数间的关系,研究其解的存在唯一性,并对给定的特殊形式的系数函数的Riccati微分方程进行了求解,给出了通解表达式.

Although the general form of Riccati differential equations is simple,there is no elementary solution.Due to its wide application,the specific solution of this equation still has research significance.According to the characteristics of the coefficient function of the equation and the concept of transforming the equation into a variable separation equation,this paper determines the relationship between the coefficient and the function,studies the existence and uniqueness of its solution,and solves a given special form of coefficient function Riccati differential equation,providing a general solution expression.