用可降阶的高阶微分方程,求y''-9y=0,设y'=p(x) 怎么求通解?

问题描述:

用可降阶的高阶微分方程,求y''-9y=0,设y'=p(x) 怎么求通解?

y'=p,即dy/dx=py‘’=dp/dx=dp/dy*dy/dx=pdp/dy带入方程:pdp/dy-9y=0,pdp=9ydy解得p=3y或p=-3ydy/dx=p=3y,dy/y=3dx,解得lny=3x+c同理,dy/dx=p=-3y,解得-lny=3x+c所以y=exp(3x+c)或y=exp(-3X+c)...