Question

解微分方程 y*y''-(y')^2-y^2*y'=0

Answer 1

yy''-y'^2=y^2y'

(yy''-y'^2)/y^2=y'

(y'/y)'=y'

两边积分：y'/y=y+C1

dy/dx=y^2+C1y

dy/[y(y+C1)]=dx

两边积分,

左边=1/C1∫(1/y-1/(y+C1))dy=1/C1ln|y/(y+C1)|+C2

右边=x+C2

所以1/C1ln|y/(y+C1)|=x+C2

ln|y/(y+C1)|=C1x+C2

y/(y+C1)=C2e^(C1x)

y=C1C2e^(C1x)/(1-C2e^(C1x))