求微分方程y"+2/(1-y)*(y')^2=0的通解

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求微分方程y

求微分方程y"+2/(1-y)*(y')^2=0的通解
求微分方程y"+2/(1-y)*(y')^2=0的通解

求微分方程y"+2/(1-y)*(y')^2=0的通解
令p=y'
则y"=pdp/dy
代入方程:pdp/dy+2/(1-y)*p^2=0
dp/p=2dy/(y-1)
积分:ln|p|=2ln|y-1|+C
得:p=C1(y-1)^2
dy/(y-1)^2=C1dx
积分;-1/(y-1)=C1x+C2
故y=1-1/(C1x+C2)

y"+2/(1-y)*(y')^2=0
y''/y'+2y'(1-y)=0
y''/y'=2y'/(y-1)
(lny')'=2(ln(y-1))'
lny'=2ln(y-1) +C
=ln(y-1)^2+C
=ln(y-1)^2+lnC1
=lnC1(y-1)^2
y'=C1(y-1)^2
y=C1(y-1)^3/3 +C2
=C3(y-1)^3+C2