特性Riccati微分方程解的探讨

Discussion on the Solution of Special Type Riccati Differential Equation

不是所有的常微分方程都能用初等积分法求解,对于形式上很简单的Riccati微分方程,一般就没有初等解法。由于其在微分方程的经典理论和近代科学有关分支的广泛应用,因此对特性方程的研究求解仍具有意义。1841年刘维尔通过找出方程的一个特解,用变量变换法将方程化为伯努利方程,进而求得方程的解。基于这一思想,研究了4种特型的Riccati微分方程,确定了方程中3个系数函数之间的关系,运用变量变换法,给出了相应的Riccati微分方程的求解过程,并给出了其解公式的表达式。

Not all ordinary differential equations can be solved using the elementary integration method.Generally,there is no elementary solution for Riccati differential equations that are very simple in form.Due to its wide application in the classical theory of differential equations and related branches of modern science,the research and solution of special equations still has significance.In 1841,Liouville transformed the equation into Bernoulli′s equation by finding a specific solution to the equation and using the variable transformation method,thereby obtaining the solution of the equation.Based on this idea,this paper studies four special types of Riccati differential equations,determines the relationship or relationship between the three coefficient functions in the equation,and uses the variable transformation method to give the solution process of several special types of Riccati differential equations,and gives the expression of their solution formula.