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数学高手请进,多谢!

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鲜花(150) 鸡蛋(3)
发表于 2005-4-3 19:41 | 显示全部楼层 |阅读模式
老杨团队,追求完美;客户至上,服务到位!
请教大家一道微分题,多谢!
. o5 b* V4 {% c) I- B1 p2 u) a2 m  ~& C8 ~4 q- M  k
d [(a+bx) ] /dx = -kc +s
  t' P- O6 j5 i9 s( m, }5 ?% c8 Zwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?, p; V; a) e; @0 E- W
# G3 Z6 ]. `, ]( m# Y" }& B
多谢了!" T& `/ b! W% [# L0 |  \
9 m6 v* M" ~' S! r4 w; R
[ Last edited by 醉酒当歌 on 2005-4-3 at 07:56 PM ]
鲜花(19) 鸡蛋(0)
发表于 2005-4-3 19:52 | 显示全部楼层
Originally posted by 醉酒当歌 at 2005-4-3 20:41:
7 t* a3 ]$ A+ I请教大家一道微分题,多谢!
% B  `7 j* g6 ^- r: U# a* _1 h) w% ~9 i0 O  y( c
d [(a+bx) ] /dx = -kc +s 1 v! `7 |: y# W5 A& Q( r
where: only x and c are unknown, others are all known,  requiire c = ?
. v' ^" W, U* D! |# y; T) ^7 K" K2 Y  ?" @" ~
多谢了!
& Z: N6 q3 d0 f: U4 T& c+ }" N

/ h" a6 q& E3 u+ W- hit is too easy:$ k4 [  Y! t4 [6 ?+ f

  n6 \& m: ?5 [$ w: l, u8 l3 }d(a+bx)/dx = b( F- \3 R4 l0 H/ s

1 P: m! G5 q( ?so
2 y& N9 m; ~% I* I( R/ k/ f
. p) F# C8 p. Jb=-kc+s' g3 ]7 R, P8 j" k
7 L, \8 t- k* d2 H+ m7 |! D
finally:  c= (s-b)/k
鲜花(150) 鸡蛋(3)
 楼主| 发表于 2005-4-3 19:57 | 显示全部楼层
Originally posted by 十年移民路 at 2005-4-3 07:52 PM:2 {% n2 p3 z; i
it is too easy:
& U0 e  s$ e5 W* \5 ^5 Y& o  ]! A
" D0 p) e8 e$ U6 X& s% id(a+bx)/dx = b
; ?7 |4 [/ _" }# l+ p" m
4 {# ]( q* H. M. f! [9 t6 sso& f1 G6 X9 R" u1 R% T

4 J  F  n/ j, T, k9 D' Cb=-kc+s
+ U# Q- W+ V* d. U, S/ `0 k& S: f4 r$ m5 V0 I
finally:  c= (s-b)/k

- _4 r# {' V$ I4 G& Y& k' S" A! g: k# l" S# a: J
the right side cannnot contain c, but b contains c, so it is still not solved.
鲜花(19) 鸡蛋(0)
发表于 2005-4-3 20:51 | 显示全部楼层
老杨团队 追求完美
Originally posted by 醉酒当歌 at 2005-4-3 20:41:4 M" e$ P$ u3 p# M: _
请教大家一道微分题,多谢!* g: f  S  U5 `- e+ j
5 I9 a5 n. G: e6 x0 \0 S# J2 Q
d [(a+bx) ] /dx = -kc +s
+ d( F0 H/ b6 [5 O+ wwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?" {' u$ F" }- f7 Z0 w5 U

9 J- \. |9 z' X! X" I多谢了!6 D4 ?* p$ d# t9 y+ b7 S, p& n" Z/ c

( K( V5 H& O, T& R& d3 T% S[ Last edited by 醉酒 ...

9 P: J7 [1 E$ }, |4 b7 C# g
5 }- \: b5 G2 T& B. K3 g9 dsorry, did you post another question? your statement is still not clear.; x. Y* U1 I% r- h, ~) t. `! Z

, ?' Q" \# R( @) h$ [  j. |2 ba, b, s, k are  constants? c means c(x) ?
5 G8 }* T" k, }7 _! s3 [4 Q/ e' \& J3 F5 |
please tell me.
鲜花(150) 鸡蛋(3)
 楼主| 发表于 2005-4-4 11:30 | 显示全部楼层
Originally posted by 十年移民路 at 2005-4-3 08:51 PM:; o* \0 Y, N- U: M
sorry, did you post another question? your statement is still not clear.
* F' J! y' r/ t* n2 W" ^/ I0 x; d+ F* r
a, b, s, k are  constants? c means c(x) ?$ x+ S/ [. ^! I3 r

& V! ^8 p) I8 f$ aplease tell me.

1 j/ _7 |8 V! g对不起,写错了,忘了变量c,应该是
  B3 s6 [5 q- q) R. r, h9 Td [(a+bx) c] /dx = -kc +s 5 f$ P0 C' k0 K) x/ O$ P5 ^$ N
, v- Y- _+ Y9 h
多谢
鲜花(19) 鸡蛋(0)
发表于 2005-4-4 18:36 | 显示全部楼层
Originally posted by 醉酒当歌 at 2005-4-4 12:30:
$ d: V$ G$ W) N- Q2 M+ [对不起,写错了,忘了变量c,应该是
7 V' e* ~) y4 \! Td [(a+bx) c] /dx = -kc +s
' {! F, L9 M7 y" I, }, {% x/ G9 a! ~2 ?9 m* ^% R4 y
多谢
% v( \4 x  }  a: t
3 X, L. k& D! c  p
do you mean it is:
& e- x' @  M: P0 u" v- Z  }7 d* `) H$ X( P  a* U  b9 N
d { (a+bx) C(x)}/dx = -k C(x) + s- C- O0 G& S+ }- p
  a" K( S6 K% e
where a, b, s are constants, you want to know what is C(x) ?; ~: ^/ R% b) r  s! ^* l

3 _' W* t' J4 a1 C8 Lif so, this needs to solve the differential equation. please comfirm it first.
鲜花(150) 鸡蛋(3)
 楼主| 发表于 2005-4-4 18:43 | 显示全部楼层
老杨团队,追求完美;客户至上,服务到位!
Originally posted by 十年移民路 at 2005-4-4 06:36 PM:: x( J8 w+ D3 k1 [
do you mean it is:$ ^% R6 R  ^, J& G1 p- f2 L
, O; o% P6 ], @8 ~# m
d { (a+bx) C(x)}/dx = -k C(x) + s
0 w5 m2 O  a% Q: {% k. V; c2 Y# |0 y6 U
where a, b, s are constants, you want to know what is C(x) ?( h0 x) {) [: v1 L% T5 i3 j
( o, _# o& L( i4 q8 y& S: L- ?
if so, this needs to solve the differential equation. please comfirm it fi ...
9 z. c4 p2 J$ A, B, G4 G- B8 b( ~

; L  e) B+ k, T+ Y. U/ r: x4 E多谢你,你说对了,就是说用x的函数表示c. 也就是 c(x), 多谢你了。请帮忙。
鲜花(19) 鸡蛋(0)
发表于 2005-4-4 19:17 | 显示全部楼层
answer:
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C(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)
鲜花(19) 鸡蛋(0)
发表于 2005-4-4 22:26 | 显示全部楼层
this answer is the good one.
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procedure:; K' H; y5 C7 [+ W6 u% E. c
  @8 t; r5 @( j9 O5 G+ H9 t& r
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s, L; S+ {+ I+ x
so:
7 o+ x* A7 I1 ^: I. [9 q
( @9 {7 d3 o& A: t* \bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
! r9 n* O5 H9 |& T$ l. w4 Ni.e.
9 O; P$ \& F7 m& I$ ?8 F  y& P! F+ {5 R' v. V
(a+bx) dC(x)/dx  = -(k+b)C(x) +s
7 g" U) ~1 ]& ^: Z& o  J4 H9 R4 S4 |

% i" [5 A5 x, N: u5 L* D/ Zintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b) . m0 {6 U5 }/ i% F( j
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx/ n9 R  o% Y1 A
therefore:
) K  ~: k- \+ `
! [1 {3 t1 y1 X1 s$ }- |( u{(a+bx)/K} dY(x)/dx=Y(x)5 ^' D' T1 t2 W' c( G" h
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from here, we can get:; }- U: ^# ], @" ]
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dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)
& a  v6 Q0 g2 S! c- J* s+ z, i# d+ d: Y6 I# R7 a/ T3 b
so that:   ln Y(x) =( K/b) ln(a+bx)
6 E; U) h% B. [4 H' E, Y% c
9 h6 `* {1 C2 B+ b2 Tthis means:  Y(x) = (a+bx)^(K/b)
; u+ p5 F" Z2 B; z( _by using early transform, we can have:6 ~) x0 C/ Y1 o+ x

* @* N# Y1 ]  {0 _-(k+b)C(x)+s = (a+bx)^(k/b+1)  U8 k" }. H/ M3 C
; |. S: N8 P* O- J1 L% d
finally:
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) h8 g2 b$ V5 o' T% u4 J6 ?C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)
鲜花(150) 鸡蛋(3)
 楼主| 发表于 2005-4-5 13:33 | 显示全部楼层
老杨团队,追求完美;客户至上,服务到位!
果然是高手,多谢了,打印出来好好研究一下。高等数学学得实在不好。你一定是工程师吧,而且数学相当好。再谢!
理袁律师事务所
鲜花(19) 鸡蛋(0)
发表于 2005-4-5 20:41 | 显示全部楼层
Originally posted by 醉酒当歌 at 2005-4-5 14:33:  o' K3 W* s7 U( X. t3 x1 U# t
果然是高手,多谢了,打印出来好好研究一下。高等数学学得实在不好。你一定是工程师吧,而且数学相当好。再谢!

: |6 w* e6 R5 ?3 H8 A9 F9 x% H0 J/ l
请你再仔细检查一遍小地方,但我可以保证方法和过程是没有问题的。6 ~4 B$ T2 P) o- x  ?
是的,我现在是注册工程师,在上大学时,曾把“基米多维奇”和“樊映川”两本习题集通做一遍。(共大约一万多题吧)。这种题真的不是难题。
鲜花(0) 鸡蛋(0)
发表于 2005-4-6 18:34 | 显示全部楼层
9楼的答案有点问题吧。  |2 A3 i+ ^+ N6 H: S& {
您说:0 q0 M, n0 d  f1 O. l( n
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d{(a+bx)*C(x)}/dx =-k C(x) + s
2 [+ v& x7 F+ B* v1 Sso:
1 c2 ?; H( |; f6 g3 k0 ybC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
5 \% t7 d4 g) P" z# A7 z4 l. E# P% l( M( ^% W/ |! M
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
4 w  U1 [% h7 N! {# ~' j但这是不正确的3 L/ J1 r0 |7 P! g9 ]0 I8 L3 c
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
9 ]4 ^9 _2 x) w# \9 R6 \6 r并不是   (a+bx)*C(x)   的导数除以   x   的导数。! ^; U: I+ O$ W5 {0 \9 |  M; W
比如说   dx/dy = (x'y-xy')/y2
3 a( O9 c; F$ n, B! u6 yx' 意思是x的导数,y'意思是y的导数。分母是y的平方。
8 o7 e4 x, |) u, I4 x+ l+ ^/ E6 Y' B因此:& x9 z1 z; E/ W; ~5 B5 t; R% x* M- _6 p, P
- u- M7 _9 y& h) v5 R$ ?
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
4 [& |$ ?7 X8 C7 p* j最右边是x的平方。7 m' g. [3 n7 H$ ]1 {' N  E
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所以您解的有问题。
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1 j8 [1 P8 J( j  Q[ Last edited by 龙舞九天 on 2005-4-6 at 07:35 PM ]
鲜花(19) 鸡蛋(0)
发表于 2005-4-6 18:53 | 显示全部楼层
同言同羽 置业良晨
Originally posted by 龙舞九天 at 2005-4-6 19:34:
1 n4 ~6 E. i" c! C; j1 A9楼的答案有点问题吧。
4 u1 G8 ?/ M( I2 E8 F8 U2 B* u/ `) G/ g您说:+ S8 X  Z! p9 p3 |

. ~: I4 s" w8 d5 u1 C$ f5 j( gd{(a+bx)*C(x)}/dx =-k C(x) + s
+ e& y0 `% x' i1 X& y% Q' sso:/ O1 P4 g% V# N2 z2 w
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s7 ~1 c) Z# [$ t3 K, B+ m

& a/ a$ p$ f$ b5 ?/ _也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
. A6 N5 m2 t0 S% F, u) y5 t3 v7 f但这是不正确的
: [& C  A3 R" q# Zd{(a+bx)* ...

5 P1 e" ~4 W9 q7 ^  C- U/ B6 z( [  [2 K" j7 o
Please note here:8 [+ G0 R8 d& h/ a& b

* j! q9 M5 Q% kd{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)3 u* Q, M" X' H, R. H
it is a multiple, not division. Right ?
鲜花(19) 鸡蛋(0)
发表于 2005-4-6 18:56 | 显示全部楼层
老杨团队,追求完美;客户至上,服务到位!
但这是不正确的6 a1 C+ `$ ]& b2 G, Y3 n+ s& p% Y6 v; E
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数# P$ W$ r' [4 Y: R
并不是   (a+bx)*C(x)   的导数除以   x   的导数。
5 x* ]3 k2 M, v& [/ I3 x3 e4 N) ?) G" z" i
no "x" in that position.
鲜花(0) 鸡蛋(0)
发表于 2005-4-6 19:30 | 显示全部楼层
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数1 `7 |" K9 O% a
....
$ ~$ G- z) t9 _, nd{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
/ Y2 K6 m+ V7 J1 ~

, w/ ~. Q0 T; c- |这样算混淆了微分和导数的概念。$ k( Q+ R* ^+ y. ?0 S- U6 q; D

3 _2 s0 ^7 b" Y3 E, m: I: Jd(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算0 l" Q; B- Y/ R! [$ Q9 v5 Z5 f
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If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h'(x) = [xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2 (应该对吧, 我也没验算)0 k3 j! T8 Y$ F
0 b. _+ U. P$ N
[ Last edited by gnome-kde on 2005-4-6 at 08:31 PM ]
鲜花(19) 鸡蛋(0)
发表于 2005-4-6 19:45 | 显示全部楼层
Originally posted by gnome-kde at 2005-4-6 20:30:
$ O4 n! ~- ?5 y6 Z% j8 Q: F这样算混淆了微分和导数的概念。' x0 v- z2 B- W3 F5 N8 m/ a

2 D1 l, o- G% F9 O! h0 q3 v& y3 jd(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算& b% N. f7 X$ z4 M

8 F9 z4 |4 a7 A2 v+ u1 FIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

1 }9 A7 l2 m4 L7 @) H, v! w' R
这样算混淆了微分和导数的概念。
: z6 O& I% g7 z' x; u: p
- ]5 h  l* e8 ~& P7 pd(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算
: ?" ]8 K! a4 e' g1 y$ n3 ?: `$ ?* P+ K; ]- u
Ah, you made a mistake: it should be:/ f7 S1 z/ |: \0 i
# @: x' h' Z' x; i: I- E" [2 {
d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
9 D7 R' I' ~; K, O/ x  V
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If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h'(x) = [xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2 (应该对吧, 我也没验算)( l3 f' O) [# B& ]" F6 H
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for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:, A2 z( I$ A/ Z  C
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h(x)=(a+bx)*C(x)
鲜花(0) 鸡蛋(0)
发表于 2005-4-6 19:51 | 显示全部楼层
同言同羽 置业良晨
Originally posted by 十年移民路 at 2005-4-6 08:45 PM:4 A7 p/ [0 f) J3 l. i- G7 [
这样算混淆了微分和导数的概念。- Z) w$ E6 A/ ?2 q+ R

3 V  ]& \  P. ^0 Y& Fd(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算0 M0 Z9 \3 D0 w7 L" L
3 K  j! ?* T+ j! e: [7 s
Ah, you made a mistake: it should be:& [0 i9 V) y" g; a: F
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d(f(x)g(x)) ...

) M/ p4 t2 o% q) |9 `
. W( U0 i$ G# ]Yes, there should not be " ' " after "f" and "g", it is a typo. :D
. |% b6 n, s& T$ p7 r  h2 t. m) S) C9 t. P. q4 K2 F
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.
鲜花(19) 鸡蛋(0)
发表于 2005-4-6 20:05 | 显示全部楼层
Originally posted by gnome-kde at 2005-4-6 20:51:6 {1 [9 r9 n2 R* m
Yes, there should not be " ' " after "f" and "g", it is a typo. :D/ C0 e: H6 x& \+ Q
, ^  c  H) @; n! F
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.
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Thanks, I am happy here to discuss math.
鲜花(150) 鸡蛋(3)
 楼主| 发表于 2005-4-6 22:24 | 显示全部楼层
Originally posted by 十年移民路 at 2005-4-5 08:41 PM:& n* X5 M3 E/ j* h9 V

5 w- y  Q* E& y+ X$ p请你再仔细检查一遍小地方,但我可以保证方法和过程是没有问题的。
7 C; C4 E; S( k9 F" j: w是的,我现在是注册工程师,在上大学时,曾把“基米多维奇”和“樊映川”两本习题集通做一遍。(共大约一万多题吧)。这种题真的 ...
, E' b7 b3 R9 S% t6 C- `
6 e2 n$ J4 B" S( D/ q
佩服!是不是微分方程如果不能transform就不能求解了,有没有现成的微分方程求解公式?
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发表于 2005-4-7 20:44 | 显示全部楼层
老杨团队,追求完美;客户至上,服务到位!
Originally posted by 醉酒当歌 at 2005-4-6 23:24:
6 L$ P( ^) _3 h7 c/ a佩服!是不是微分方程如果不能transform就不能求解了,有没有现成的微分方程求解公式?

% F0 i/ N& n: E- r7 y
$ X; m3 L' a( \9 L  f/ z3 ]: Y2 M理论上,只有很少的几类”齐次常微分方程“可以写出其”封闭的解析解“。不作变换,微分方程是非齐次的,可能不太容易作到。一般地说,没有什么现成的微分方程求解公式。但对”线性定常齐次微分方程“是有一个较方便的解法。
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