一阶常微分方程通解中任意常数的分析及教学

The Analysis and Teaching of the Arbitrary Constant in the Usual Solution of First Order Ordinary Differential Equation

摘要: 本文主要从两个方面分析一阶常微分方程通解中任意常数的含义。一：任意常数不一定取遍所有的实数；二：任意常数和方程的特解不一定是一一对应的，即取定一个任意常数不一定仅仅只确定一个特解，也不是每个特解都可以由通解中的任意常数可以确定。最后为方便使用常数变易法求解非齐次一阶线性微分方程的通解，举例阐明了一阶线性微分方程对应齐次方程通解的规范形式。

Abstract: The arbitrary constant of the usual solution to first order ordinary differential equation is discussed in this paper. Firstly, the arbitrary constant in the usual solution does not necessarily take over all of the real numbers. Secondly, the arbitrary constant and special solution is not necessarily a one-to-one correspondence, that is, a constant does not determine a unique special solution, also is not that each special solution can be determined by the arbitrary constant in usual solution. In order to use the method of variation of constant to solve the first order linear differential equation, the canonical form of the usual solution to the homogeneous equation of first order linear differential equation is provided.