求微分方程满足初始条件的特解:y''=e^2y,y(0)=y'(0)=0
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求微分方程满足初始条件的特解:y''=e^2y,y(0)=y'(0)=0
令p=y'=dy/dt,那么有:
y''=dp/dt=(dp/dy)(dy/dt)=pdp/dy
将上述结果代入原方程得到:
p(dp/dy)=exp(2y)
分离变量得到:
pdp=exp(2y)dy
等式两侧取不定积分得到:
p²/2=[exp(2y)]/2+M···········································M为任意常数
整理得到:y'=dy/dt=p=√[exp(2y)+N]·····································N为任意常数,N=2M
代入初始条件解得:N=-1
再次分离变量得到:
dy/√[exp(2y)-1]=dt····················································※·
令y=lnθ,即θ=exp(y),等式两边取微分得到:dy=dθ/θ
将以上结果代入方程※,得到:
dθ/[θ√(θ²-1)]=dt······················································✿
利用换元积分法,令θ=secδ,同理等式两边取微分得到:dθ=secδtanδdδ,代入方程✿,得到:
dδ=dt
两边积分得到:
δ=t+L···········································L为任意常数
所以有:
exp(y)=θ=secδ=sec(t+L)
代入初始条件得到:L=0
所以有题目方程的特exp(y)=sect 或者y=ln|sect|
y''=dp/dt=(dp/dy)(dy/dt)=pdp/dy
将上述结果代入原方程得到:
p(dp/dy)=exp(2y)
分离变量得到:
pdp=exp(2y)dy
等式两侧取不定积分得到:
p²/2=[exp(2y)]/2+M···········································M为任意常数
整理得到:y'=dy/dt=p=√[exp(2y)+N]·····································N为任意常数,N=2M
代入初始条件解得:N=-1
再次分离变量得到:
dy/√[exp(2y)-1]=dt····················································※·
令y=lnθ,即θ=exp(y),等式两边取微分得到:dy=dθ/θ
将以上结果代入方程※,得到:
dθ/[θ√(θ²-1)]=dt······················································✿
利用换元积分法,令θ=secδ,同理等式两边取微分得到:dθ=secδtanδdδ,代入方程✿,得到:
dδ=dt
两边积分得到:
δ=t+L···········································L为任意常数
所以有:
exp(y)=θ=secδ=sec(t+L)
代入初始条件得到:L=0
所以有题目方程的特exp(y)=sect 或者y=ln|sect|
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