作者LuisSantos ( )
看板trans_math
標題Re: [考古] 高大96年
時間Tue Jul 1 20:42:35 2008
※ 引述《reideen107 (理想一直都在)》之銘言：
: 3.The sum of two nonnegative numbers is 20. Find the numbers
: (a)if the product of one number and the square root of the other is to be
: as large as possible;
: (b)if one number plus the square root of the other is to be as large as
: possible.
: 我看不懂這題在說啥@@，請問要怎麻解？
: 4
: 6. (a) Evaluate ∫ log x /x dx.
: 1 2
: (b) Solve the equation
: dy
: x— = x^2 + 3y
: dx
: given the initial condition y(1)=2
: (a)題的log是要直接化成lnx/ln2去積嗎？
: (b)題不太懂......
4
(a) ∫ (log x)/x dx
1 2
4 lnx 1
= ∫ (-----)(---) dx
1 ln2 x
1 4 lnx
= (-----)(∫ ----- dx)
ln2 1 x
1 (lnx)^2 |4
= (-----)(---------) |
ln2 2 |1
(ln4)^2
= -------
2ln2
((2)(ln2))^2
= ------------- = (2)(ln2)
2ln2
(b)常微分方程的積分因子法
dy
(x)(----) = x^2 + 3y
dx
dy 3
---- = x + (---)(y)
dx x
dy 3
---- - (---)(y) = x
dx x
積分因子 = e^(∫(-3/x) dx)
= e^(-3lnx)
= e^(ln(x^(-3))) = x^(-3)
dy 3
---- - (---)(y) = x 的等號兩邊同乘以 x^(-3) 得
dx x
dy
(x^(-3))(----) + (-3)(x^(-4))(y) = x^(-2)
dx
d((x^(-3))(y))
---------------- = x^(-2)
dx
-1
(x^(-3))(y) = ---- + c => y = -x^2 + (c)(x^3) , y(1) = 2 代入
x
2 = -1 + c => c = 3
y = -x^2 + (3)(x^3)
: 題目都沒答案，寫完都不知道對不對@@
: 煩請各位高手相助~QQ
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◆ From: 61.66.173.21
1F：推 reideen107:(a)應該是2ln2吧？ 1/2不見了@@..122.121.201.135 07/01 21:22
2F：→ reideen107:(b)看起來頗複雜...感謝大大的解題122.121.201.135 07/01 21:23
※ 編輯: LuisSantos 來自: 61.66.173.21 (07/01 21:34)