let f(x)=x//x/ for x不等於0, k for x=0 where is constant show that no matter how the constant k is chosen the differential equation y'=f(x)has no solution on an interval containing the origin.
f '(x)=[x→0]lim[(f(x)-f(0))/(x-0)]
=[x→0]lim[((x/lxl)-k)/x]
=[x→0]lim[(1/lxl)-(k/x)]
1.當[x→0+]lim[(1/lxl)-(k/x)]=[x→0+]lim[(1/x)-(k/x)]
=[x→0+]lim[((1-k)/x)]=∞
2.當[x→0-]lim[(1/lxl)-(k/x)]=[x→0-]lim[(-1/x)-(k/x)]
=[x→0-]lim[((-1-k)/x)]=∞
故函數f(x)在 x=0,不可微分
所以the differential equation y'=f(x)has no solution on an interval containing the origin.
y'=f(x)has no classical solution on an interval containing the origin. Some weak solutions still can be defined.
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f '(x)=[x→0]lim[(f(x)-f(0))/(x-0)]
=[x→0]lim[((x/lxl)-k)/x]
=[x→0]lim[(1/lxl)-(k/x)]
1.當[x→0+]lim[(1/lxl)-(k/x)]=[x→0+]lim[(1/x)-(k/x)]
=[x→0+]lim[((1-k)/x)]=∞
2.當[x→0-]lim[(1/lxl)-(k/x)]=[x→0-]lim[(-1/x)-(k/x)]
=[x→0-]lim[((-1-k)/x)]=∞
故函數f(x)在 x=0,不可微分
所以the differential equation y'=f(x)has no solution on an interval containing the origin.
y'=f(x)has no classical solution on an interval containing the origin. Some weak solutions still can be defined.