y''=(y')^3+y'求解微分方程的通解答案是sin(y+C1)=C2e^x求两种方法T.T

Question

y''=(y')^3+y' 求解微分方程的通解
答案是sin(y+C1)=C2e^x
求两种方法T.T

Answer 1

设p=y'
则y"=dp/dx=dp/dy* dy/dx=pdp/dy
方程化为：pdp/dy=p^3+p
dp/dy=p^2+1
dp/(p^2+1)=dy
arctanp=y+c
p=tan(y+c)
dy/dx=tan(y+c)
dy/tan(y+c)=dx
cos(y+c)dy/sin(y+c)=dx
d(sin(y+c))/sin(y+c)=dx
ln[sin(y+c)]=x+c1
sin(y+c)=c2e^x