如图,可降价的高阶微分方程

Question

如图,可降价的高阶微分方程

Answer 1

令p=dy/dx,则d^2y/dx^2=pdp/dy
代入原式得：2ypdp/dy=p^2+y^2
2dp/dy=p/y+y/p
再令u=p/y,则p=yu,dp/dy=u+yu'
2(u+yu')=u+1/u
2yu'=-u+1/u
2du/(-u+1/u)=dy/y
2udu/(1-u^2)=dy/y
[1/(1-u)-1/(1+u)]du=dy/y
积分：-ln|1-u|-ln|1+u|=ln|y|+C1
得1-u^2=C/y
1-p^2/y^2=C/y
y^2-p^2=Cy,将p(0)=-1,y(0)=1代入得;1-(-1)^2=C,得c=0
因此y^2-p^2=0,
得p=±y
dy/dx=±y
dy/y=±dx
积分：ln|y|=±x+C2
得y=Ce^(±x)
代入y(0)=1,得：1=C,
因此y=e^(±x)