Question

为什么这个是微分方程的解？

概念有点不太明白求讲解

Answer 1

y1,y2是方程y'+p(x)y=q(x)的两个解，有
y1'+p(x)y1=q(x)
y2'+p(x)y2=q(x)
两式相减得
(y1'-y2')+p(x)(y1-y2)=0
(y1-y2)'+p(x)(y1-y2)=0
或(y2-y1)'+p(x)(y2-y1)=0
即y2-y1是方程y'+p(x)y=0的解
同理
两式相加得
(y1'+y2')/2+p(x)*(y1+y2)/2=q(x)
即(y1+y2)'/2+p(x)*(y1+y2)/2=q(x)
(y1+y2)/2是方程y'+p(x)y=q(x)的解