Re: [微積] existence and uniqueness
題目是要問有沒有存在唯一嗎? 且解存在的範圍?
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<Theorem> (Existence and Uniqueness)
if f：D → R^n , D open in R^(n+1) , f€C^1(D) (其實不用那麼強 不過你這題符合)
then for all (t_0,x_0)€D, the initial value problem {x'(t)=f(t,x(t))
{x(t_0)=x_0
has unique maximal solution defined on an open interval. That is,
(1) there exists a solution s(t) defined on (a,b) (a maybe -inf, b maybe +inf)
such that if p(t) is a solution defined on I (an interval in R)
then I is included in (a,b)
(2) if p(t), q(t) are two solutions defined on I, J respectively
then p(t)=q(t) on the intersection of I and J
Moreover, if D=R^(n+1) , f is bounded function (│f│<= M )
then (a,b) = (-inf, +inf) = R
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所以你這題的f(t,x)=sin(tx) , 顯然C^1(R^2)
所以存在唯一解在整個R
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04/13 15:12, , 1F
04/13 15:12, 1F
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