Question

设y1y2是某个二阶线性常系数齐次方程的两个解

8．设y1(x),y2(x)为二阶常系数齐次线性方程y"+py'+qy=0的两个特解,则c1y1(x)+c2y2(x)(c1,c2为任意常数)是该方程通解的充分必要条件是
(A) y1(x)y'2(x)-y2(x)y'1(x)=0． (B) y1(x)y'2(x)-y2(x)y'1(x)≠0．
(C) y1(x)y'2(x)+y2(x)y'1(x)=0． (D) y1(x)y'2(x)+y2(x)y'1(x)≠0．

Answer 1

简单分析一下，答案如图所示

Answer 2

A选项
根据齐次线性方程的解的性质,y1(x)≠ky2(x)时,c1y1(x)+c2y2(x)是方程的通解.
令F(x)=y2(x)／y1(x),F′(x)≠0时,上式成立.
即 y′2(x)y1(x)-y2(x)y′1(x)／[y1(x)]²≠0
∴选A