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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!& X* |6 A, E) A" h2 ~" a: L, r; o

0 N& N% o% k% gd [(a+bx) ] /dx = -kc +s
, w; U  z/ E$ X( n3 iwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?' O/ F+ h, b9 k  h# W+ u; }

- B" p- W3 e: A- D2 l% w多谢了!9 z) M9 D5 x' E
% R) F5 h* }" f7 g+ n: t
[ 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:$ a5 H/ e/ w' _2 a' c
请教大家一道微分题,多谢!
, a- q- ?2 b1 n0 p! U  G' f8 L% x& ^5 \
d [(a+bx) ] /dx = -kc +s
( Q* n& Z: I3 r1 ^where: only x and c are unknown, others are all known,  requiire c = ?
8 A6 a0 [0 }* E2 r
6 V2 f0 f# o3 c; t# q$ [# t7 z多谢了!

  @) L" k6 r  K  i' z
* Q% e* g2 k% ]1 xit is too easy:
$ X( W% W. d- ~( ?9 y
5 D9 M6 O  ~0 [9 M+ ~, n& yd(a+bx)/dx = b% ?) L+ o- i8 X6 p4 G) T+ M7 P$ d9 U

( G* E- E; N( d. t: H3 g4 Mso6 l0 F4 e% k* |) g- U7 s% F
! q" `8 @/ ^3 d/ e9 Z: n* k- n5 i
b=-kc+s
( P0 J1 p4 o7 p5 E
- C- V7 h7 N* K; _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:* l4 h. i& O$ W( [7 N0 O. Q2 p/ V8 k& F
it is too easy:
) o$ O! S6 K' g- n) T7 U7 Q& b  A& J  ]6 F& W. Z- B) Y
d(a+bx)/dx = b. w+ y' u4 g& O# |2 h' ?

7 H: p/ ?- \2 f5 Y  D" Lso( Y. \6 P* {2 u" }! x, v; Q
+ [" ^- m( E7 O+ t
b=-kc+s- z8 n6 C4 q! q5 v- L$ i
* w9 |2 i9 |; X/ ~& I) z
finally:  c= (s-b)/k

# c2 R* Z& i2 `2 }1 {
. G, ^3 }5 Y. S* H& \1 H; a( ~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 20:41:1 |  V& I& K6 ^9 [
请教大家一道微分题,多谢!
1 |! _: P/ l( [0 Q' `- N( C
7 P6 {. s  Z& A: {9 J; n( cd [(a+bx) ] /dx = -kc +s
& ^, G0 [! v3 Qwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?+ i; b- B8 Q6 e9 b0 j8 b
+ }% h: j) R, @7 `9 `
多谢了!5 I4 G, B1 @  `3 m. T

# i! k- ~0 O- B$ j$ _" c[ Last edited by 醉酒 ...
$ g% s1 N# |1 _* h( S  I

$ _0 K5 Y9 n1 y& o) ^9 fsorry, did you post another question? your statement is still not clear.
8 n  ?$ a1 `3 n# B+ X. A) m7 _2 q$ ]* w+ t/ ?
a, b, s, k are  constants? c means c(x) ?7 N+ f6 M* I$ I' i8 t, d
7 C/ n  f. u3 x
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:
$ ?# B5 d8 v6 v  B% l7 Isorry, did you post another question? your statement is still not clear.3 u- f' K$ t& p
3 n9 U6 ]" R+ U- ~7 W2 Q
a, b, s, k are  constants? c means c(x) ?* F8 u' }$ i0 c" v" S
- R) D2 b, I* X4 ]  ~
please tell me.

- L" U( r5 _; ~; ~' h对不起,写错了,忘了变量c,应该是
, f- I: c) U1 x* x6 p; t8 q; W+ j. L5 W' Od [(a+bx) c] /dx = -kc +s
) _- M  e' O6 {' f# K  A7 ~. L4 l7 P
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 12:30:0 ?5 J, E: z* V
对不起,写错了,忘了变量c,应该是
! w( |- l2 \* v" Rd [(a+bx) c] /dx = -kc +s $ I! p4 b; D- P% Z2 X! f
) _" M% ]& |* r$ {
多谢

7 c: `* Q* F0 |4 Y3 Q: v) M0 p( r: y; I0 b% E
do you mean it is:
+ o) J2 L) g1 E$ E% A  ^
4 z: }( P5 u' @4 Gd { (a+bx) C(x)}/dx = -k C(x) + s2 B1 ^" h( `7 z) Q  ~, Y7 D

' b; L( W. y4 T2 g& \where a, b, s are constants, you want to know what is C(x) ?( K" N* b- Z4 _$ ?% _. @$ T

  e: W7 Q5 V5 k' W7 hif 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:
' F$ U* Z$ f+ Rdo you mean it is:+ k+ C) t0 d7 r8 E- [3 G0 ~5 X
9 O7 D" }$ F) z/ _( @# B
d { (a+bx) C(x)}/dx = -k C(x) + s! C8 e# D0 Q: ~! G4 n

; m. {3 g. V5 wwhere a, b, s are constants, you want to know what is C(x) ?9 S( u- w1 _2 ~+ W" D  F1 G& l

1 }: b5 o" @; I: h$ c1 }, U# Zif so, this needs to solve the differential equation. please comfirm it fi ...
+ ^, Q. l2 _! [4 t/ ?$ t
3 H' ^5 ]* ]" I! ?
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.- h6 h0 V  U' [% j. K% g

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( q+ S/ M! D  F" k1 r9 Y/ wprocedure:
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+ t3 Y8 d2 J) i  B4 H) xFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s! {' T& S# ^) R. o/ P6 a& e
so:, M' p5 z! u2 E3 v$ `) j$ Q" b
4 Q0 s8 t: R7 ?( l- _! ]" Q! I
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
8 a1 K1 H+ y- _; H, `9 ~i.e.
8 B; t, j5 H# E, h6 M0 X: V/ v
) o6 V3 T* m* {: O+ h2 B(a+bx) dC(x)/dx  = -(k+b)C(x) +s1 B# S* n$ Y9 {& W$ K
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introduce a tranform: KC(x)+s =Y(x), where K=-(k+b) 0 C9 e; i5 d/ c+ P% D9 L* w
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx2 P6 P; ~  R; ?7 ^& O
therefore:
, ^5 t( L2 a2 v& h7 Y
: Y) t$ Q$ B% T; F1 ~{(a+bx)/K} dY(x)/dx=Y(x)9 }2 b2 _5 y; o
) m: H% i' A, ]' W4 l
from here, we can get:# S: o) F, }7 S" o
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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)9 y3 j5 t; i* W# ^1 _
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so that:   ln Y(x) =( K/b) ln(a+bx)6 v) s3 }  h% Y# U7 ~! @7 a
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this means:  Y(x) = (a+bx)^(K/b), ?2 Y! [& I0 C
by using early transform, we can have:
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-(k+b)C(x)+s = (a+bx)^(k/b+1)
# @+ r6 R; \6 H1 G( Y7 I$ m2 v7 R0 e$ y& L# i
finally:4 e+ o/ A! b% j7 Z
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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!
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:
4 E/ n4 Q0 j0 r0 G: J( E% y$ Q) ~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!
0 G7 |6 E: f+ Z: G' S4 s
( I. _$ R2 b  m# p' q$ Q: v1 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。
. E$ E  o1 x+ G7 H* D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。6 ?( @. S$ t4 @) W% d8 A/ n
您说:0 p" e! r4 R. G
+ Q1 ^& d  x; r' L0 B3 m
d{(a+bx)*C(x)}/dx =-k C(x) + s
  k4 ~+ y1 [* u, Y8 yso:: o3 u9 D! v8 r% R# W1 \2 f1 k
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
& u, q1 ^+ w( o! y; V, t. x$ V% V. E7 x9 \( `- H9 d
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx& Q( o! L, M% a5 \% y. @5 s
但这是不正确的
& ?8 i( E& D" C, w/ Ud{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
  @9 ]# c1 l' ]) B, c4 n! Q并不是   (a+bx)*C(x)   的导数除以   x   的导数。( W4 n) m' w1 ], b: W- F/ M
比如说   dx/dy = (x'y-xy')/y2
7 B4 F: {' s/ L5 H" ^; Wx' 意思是x的导数,y'意思是y的导数。分母是y的平方。) }5 D  ^- v/ I9 G! z9 r
因此:
6 H5 j* V% p* G
( m# B5 G; a# u8 ld{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
. ?8 x$ O  s/ [  Y# P8 j$ w% j$ G( n最右边是x的平方。
( c; {- q% z4 ?; T6 a/ b6 C: U
, k. [; g4 S" k所以您解的有问题。/ @$ b$ @% K' w9 r

( H2 N1 G& A( H3 i' G: b! q9 O+ h[ Last edited by 龙舞九天 on 2005-4-6 at 07:35 PM ]
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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:% ]% g, T5 B0 N5 t/ x
9楼的答案有点问题吧。$ w$ ]1 D/ U7 L) z1 C+ o
您说:
. S: b' p( m- @' z/ n- S0 L
/ N4 b8 S% E" B) G6 Kd{(a+bx)*C(x)}/dx =-k C(x) + s2 G, i4 e9 H: l: a3 [3 q
so:( r6 M/ u5 ^8 f: Q- U2 Q- B
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s" d  w; Z) c# _  u: j* d  a! Z* U
& y1 N% |6 |; ]+ p& G
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx2 P) d4 Z9 Z' S0 j
但这是不正确的/ t1 f" [6 l0 B
d{(a+bx)* ...

( X2 u' `& I) L: s& A1 D. z/ z6 p
3 F) R4 M/ }# U& y& m  f* DPlease note here:
( k5 |8 b8 e1 }' E
- L) i" g6 j6 Z+ d/ ad{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)2 s1 N& B' A) y8 I( B
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, M5 `1 U: [& s, z4 H7 L
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
: b2 s& ^6 H; R并不是   (a+bx)*C(x)   的导数除以   x   的导数。
) {- x4 ^' }& M5 S
; A; ^( c( i, s, I, _: tno "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  t, P. R( S1 B, j" N5 h8 a. O
...." R  }9 L  w- W
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

+ ~" W5 p* d+ W% P7 C7 h6 ~6 H2 G
( b& Y0 B  Q+ e. O: |6 q. g这样算混淆了微分和导数的概念。4 W, v% D% w% Y- }3 q

, \# \$ w/ x$ h1 w5 r1 k3 O8 xd(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 M% C7 w* f7 p$ H- ?
6 b7 J3 H8 F: FIf 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)
3 a7 d/ w; z( N0 Z( p) o8 i1 r: ]% o( `5 D- ]
[ 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 20:30:0 a9 H( p, x* u4 ~; {- d
这样算混淆了微分和导数的概念。
9 d0 E! g$ `+ {7 K; M
6 d3 A- b* a/ A- 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& a+ f% h( l0 c+ t4 @3 A# `
+ ]5 a  ?9 m: }5 C: ~& R% x
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

5 ~9 M4 @2 Z* m1 ?6 B( i
" n6 w. K$ {9 c  j( p这样算混淆了微分和导数的概念。
( J9 i) W( P6 }$ ~5 U) g: j/ e. C- D9 ~1 s8 ?; P- s
d(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
6 j& U* J5 M2 B/ D( `  T
3 y+ w  T6 F0 ^Ah, you made a mistake: it should be:
8 I+ B; E$ E) [* n
. k. R0 {& p- `: b3 gd(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
' F- ]8 x$ R2 w/ [
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) B) {: f$ [% U4 V% G, j$ @
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)" m+ G& C$ }# `# q$ ~5 h8 x1 A

3 V0 i6 H/ U# U% X6 Pfor h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:6 V6 a0 k% P, ~9 i6 L

. |$ n; a6 C+ s- ~& q" th(x)=(a+bx)*C(x)
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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:) I% {: b5 K1 \
这样算混淆了微分和导数的概念。& c6 Q5 r+ e! P9 O6 s2 d

+ w9 l0 T2 U; I8 i5 z3 a  Gd(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
+ i9 J8 V0 t- f4 [
0 s2 h7 T  m* q7 l3 n2 {% |3 BAh, you made a mistake: it should be:
+ M+ d( y. P2 g" f' b
+ [$ J: }5 n! ?; `% w( td(f(x)g(x)) ...
" z' s- |; o3 Z7 F1 v8 H; \

* Q# K. Z7 W" Z% s/ MYes, there should not be " ' " after "f" and "g", it is a typo. :D
! r2 T0 T; ~, ^+ c( M/ w5 K6 t: {6 e& P& h) v' P
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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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:
+ M, t4 l7 o+ {5 XYes, there should not be " ' " after "f" and "g", it is a typo. :D$ ^( q8 {' Q$ [  S/ W) ]5 a
1 w  u$ ]3 Z  f5 Q
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.
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 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:$ W6 s6 y' u9 K, s% ^
4 m8 ~5 F7 H$ y. B' a2 @( J
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。
, _; w/ H3 m5 N4 [7 _# `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 ...
: T& h" d; |8 P5 K5 W
0 T# ?, V  @; R/ g
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:
" W5 c$ `" G- H; P% R. A$ e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?
% x1 f* I" i$ k( f6 U

% E4 Y# z: j+ V' Z* b, h5 l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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