y(t)=?
dy/dt=(1-y^2)/y ,y(0)=-2
dt/dy=y/(1-y^2), dt=[y/(1-y^2)]dy,
t=∫[y/(1-y^2)]dy=-(1/2)∫[1/(1-y^2)]d(1-y^2)
=-(1/2)ln|1-y^2|+C
y(0)=-2, 即t=0时,y=-2
0=-(1/2)ln|3|+C, C=(1/2)ln3
t=(-1/2)ln|1-y^2|+(1/2)ln3=-(1/2)ln(|1-y^2|/3)
|1-y^2|/3=e^(-2t)
得一个用方程表示的隐函数
|1-y^2|e^2t=3
考虑到符合y(0)=-2,y=-√[1+3e^(-2t)]。 全部
dy/dt=(1-y^2)/y ,y(0)=-2
dt/dy=y/(1-y^2), dt=[y/(1-y^2)]dy,
t=∫[y/(1-y^2)]dy=-(1/2)∫[1/(1-y^2)]d(1-y^2)
=-(1/2)ln|1-y^2|+C
y(0)=-2, 即t=0时,y=-2
0=-(1/2)ln|3|+C, C=(1/2)ln3
t=(-1/2)ln|1-y^2|+(1/2)ln3=-(1/2)ln(|1-y^2|/3)
|1-y^2|/3=e^(-2t)
得一个用方程表示的隐函数
|1-y^2|e^2t=3
考虑到符合y(0)=-2,y=-√[1+3e^(-2t)]。
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