Re: [微積] ODE

[Vulpix (Sebastian)]

※ 引述《oblivion87 (oblivion87)》之銘言：
: 毫無頭緒，只想出這個幾乎是亂猜的方法，請問這樣判斷可以嗎，還是要硬級數解呢，有
: 什麼比較嚴謹的證明方法嗎？
: https://i.imgur.com/3Cyv9f7.jpg

你的題目應該是：

f"+f=0, (x+1)g"+xg=0 for x>0, both f and g are not trivial,

show that there is a zero of f between any consecutive zeroes of g.


pf):

Take two consecutive zeroes of g, say a and b, where a < b.

suppose that there is no zero of f in (a,b),

then f remains its sign on (a,b), say positive.

WLOG, we may assume that g is positive on (a,b) as well.


f"g-fg" = -fg+fgx/(x+1) = -fg/(x+1)

Integrate the previous equation on (a,b), we get

    f'(b)g(b)-f(b)g'(b)-f'(a)g(a)+f(a)g'(a) = -∫_a^b fg/(x+1) dx

=>  -f(b)g'(b)+f(a)g'(a) = -∫_a^b fg/(x+1) dx

, where RHS < 0.

By definition, g'(a)≧0 and g'(b)≦0.

They cannot be 0, since g'(a) = 0 implies g = 0.

So g'(a) > 0 and g'(b) < 0, hence LHS≧0, a contradiction.

--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 163.13.112.58
※ 文章網址: https://www.ptt.cc/bbs/Math/M.1545025905.A.6AE.html
推 oblivion87  : 題目是台大應數所考古題，是問那個ode oscillate fa
→ oblivion87  : ster
→ oblivion87  : 我也覺得怪怪的，google到的比較定理
→ oblivion87  : 好像只給出zero間存在令一de的條件
→ oblivion87  : 但沒特別提到哪個ode會oscillate faster
推 oblivion87  : https://i.imgur.com/ncmC3UJ.jpg
→ Ricestone   : 題目後半可以看成它定義什麼叫oscillate faster
※ 編輯: Vulpix (61.230.70.207), 12/29/2018 12:13:13