设y1(x),y2(x)为二阶变系数齐次线性方程y’’+p(x)y’+q(x)y=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条件为

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问题 设y1(x),y2(x)为二阶变系数齐次线性方程y’’+p(x)y’+q(x)y=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条件为

选项 A、y1(x)y2’(x)一y2(x)y1’(x)=0.
B、y1(x)y2’(x)一y2(x)y1’(x)≠0·
C、y1(x)y2’(x)+y2(x)y1’(x)=0.
D、y1(x)y2’(x)+y2(x)y1’(x)≠0·

答案B

解析 根据题目的要求,y1(x)与y2(x)应该线性无关,即≠λ(常数).反之,若这个比值为常数,即y1(x)=λy2(x),那么y1’(x)=λy2’(x),利用线性代数的知识,就有y1(x)y2’(x)一y2(x)y1’(x)=0.所以,(B)成立时,y1(x),y2(x)一定线性无关,应选(B).
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