第一題已經有人做了
提供第二題一些想法
因為 1 + x^2 + xy + y^2 = 1 + 3/4 x^2 + (y+x/2)^2
所以所有的解都是嚴格遞增且 |y(x)-y(0)| < M for all x
有了 sup_n ∣f_n(1/n)∣＜∞
當然就有 sup_n {sup |f_n(x)|} 且每個f_n都是嚴格遞增這樣的條件
這樣當然可以找到一組Subsequence Converge to some f
接下來的證明就都很直接明瞭了
※ 引述《gok338 (HuaHua)》之銘言：
: 1.試判斷正確性，並證明
: There exists no function U(x,y) which is C^2 in R^2 and satisfies
: U(x,y) = 0 on x^2 + xy + y^2 =1, U(x,y)≧0 and Uxx + Uyy = 1+U^2
: for x^2 + xy +y^2＜1.
: 還有C^2 in R^2 是什麼意思?? 謝謝~
: 2.Assume that f_n: R→R (n=0.1.2...) is a sequence of differentiable
: functions s.t each f_n(x) is a solution of the equation
: y'(1 + x^2 + xy + y^2) = 1. If sup_n ∣f_n(1/n)∣＜∞,
: prove that there exists a subsequence f_n_k(x) s.t
: lim f_n_k(x) = f(x) exists for x in R.
: k→∞
: This limit f(x) must also be a solution of y'(1 + x^2 + xy + y^2) = 1.
: 謝謝~
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