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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!" ~( K, B- [0 n

" x3 \8 U7 z9 C, g% m7 W1 td [(a+bx) ] /dx = -kc +s * R# L5 k( R, D- S3 @  O! y0 O
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
  K: c! `2 L  V& d% h6 Z
( y8 y, m3 F  Y& T4 O# K/ Q多谢了!
0 ^2 G  y, q7 _: d
, `" w$ }7 x/ Q2 `4 t! e[ 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:
) T6 b1 Z, N1 }$ Y请教大家一道微分题,多谢!: `: k" C* _" \1 ?: E6 @

8 |4 Y. X8 Y/ b% L7 r5 z- td [(a+bx) ] /dx = -kc +s 0 n/ Q) \8 K. A6 W) K' O
where: only x and c are unknown, others are all known,  requiire c = ?/ O3 _( z& ]+ M* ]+ t8 ?
% t1 a/ E9 X: H+ z3 x" H
多谢了!

2 K( {# b2 h& f% z1 L+ S% _- m! }) D+ v' S  i) H! R
it is too easy:) L6 l- n- Q, q5 D$ [6 g
4 K* T: I& f7 S
d(a+bx)/dx = b! U! Q* h* k1 S7 a( _4 \/ ?
) Z+ G; t2 I8 a
so, g2 L7 s; t7 c2 ?" e% G

+ B  Z8 G, G5 T% d& lb=-kc+s, z0 g& {' T. v* h; {* X4 L' J
1 ~3 W9 Y: N. ?3 J3 a$ X
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:0 a# o* _- U7 E' J& h
it is too easy:4 @2 x; a  g1 Y& N

* Z5 ^) o% C- Td(a+bx)/dx = b2 y0 u# g# T& M, _4 H( n& k
$ O  l8 g: ~6 ]; ^4 X& s7 {
so. N& j" Z& {  T8 x' m; v2 ]& {

6 x' u6 k7 E4 L4 ~b=-kc+s' }0 m9 {$ e# x

- \( I3 e! u% K  d% H' g6 ^5 Vfinally:  c= (s-b)/k

/ v% @# p; J$ P( l( U3 ?, I1 c) Z3 i2 c! Q
the right side cannnot contain c, but b contains c, so it is still not solved.
理袁律师事务所
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:
2 f* _, ?) D6 y0 ]请教大家一道微分题,多谢!# H% A' N% |* v' z

- q3 G% p9 s* J% ^. O# g* kd [(a+bx) ] /dx = -kc +s
9 v* `1 d5 Z" z5 Gwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
3 }% d, \, j, i! J1 r
; I" x6 p) E  M( Z) E! W多谢了!
7 ^4 m% \$ j- h3 r# Y9 R& G. A! ^! G
[ Last edited by 醉酒 ...
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sorry, did you post another question? your statement is still not clear.4 G5 @# o1 \  u' {! u2 h2 R' ~6 |, d
- Y" w" l0 G3 I/ X& t% W' f
a, b, s, k are  constants? c means c(x) ?
: O# {/ X& F0 N! s$ v
; q2 C; h9 b' Y5 U/ wplease 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:
$ p4 G2 R) y: W6 Dsorry, did you post another question? your statement is still not clear.
. U6 J  ?# ~2 y; {1 E- |2 \& P1 F- w& H& ^' L
a, b, s, k are  constants? c means c(x) ?
! C, B" |( n& g) t
/ q' X7 s# A4 Qplease tell me.

7 q$ @! U6 ]! Y: ^, T, R对不起,写错了,忘了变量c,应该是
! d* @1 p' N* y. U7 ?7 td [(a+bx) c] /dx = -kc +s
8 a5 ~# e; P4 M! ]) ~, ]5 e* W8 @) L0 I! N# v/ g
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:
" f9 c0 _+ ?1 b7 ^6 s对不起,写错了,忘了变量c,应该是" }# ~, B' v: _% t
d [(a+bx) c] /dx = -kc +s   W" a! M# d% p: V3 S

  r, P' a3 k6 j" G多谢

: R! H' ?/ u: \: y# _6 C0 ^3 b% L2 F! g$ \
do you mean it is:5 ~% C7 b' V( f; |4 V

' _' v+ L0 ^3 K# ^5 Z- k/ ~6 z) gd { (a+bx) C(x)}/dx = -k C(x) + s1 d8 |0 E" @1 Q# Z- J$ U

! V( F0 P5 v; M2 _4 B3 k: twhere a, b, s are constants, you want to know what is C(x) ?
+ L! \9 _: B' c5 v" N/ v' {5 U
3 [$ e4 p* |7 F: _" cif 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:' g* g8 R' |, L' P; A% k8 s- H* G7 Y
do you mean it is:
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d { (a+bx) C(x)}/dx = -k C(x) + s& c3 P2 ?* W# s7 Z2 [/ y! J

% |: V4 B" d# |3 k! _/ w9 {where a, b, s are constants, you want to know what is C(x) ?# y, g3 s4 ^1 E9 X9 Z$ U( }: [
' D, n! u$ K' L
if so, this needs to solve the differential equation. please comfirm it fi ...

+ p( @2 X- r0 d- G
& @* }- K1 c2 l+ S6 @: B% |1 R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./ M! h* Y- l5 c+ }- T% l9 ^
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3 P% @; \" J5 }# B" O3 G/ Qprocedure:
& _4 B6 S! t9 [0 v  ]
5 s  }2 s  P2 P- DFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s0 ^3 I6 n9 v" ~4 Q1 e$ Y
so:/ X6 {4 J5 D& p+ y8 ?
, {$ U7 |' i' X9 S9 c$ e- }% k2 r+ m
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s% s: P& m6 y7 U- S* w
i.e.
  }# H/ z& Q2 V6 ~# d
  ^4 m* h5 Q6 _& s( c6 |(a+bx) dC(x)/dx  = -(k+b)C(x) +s( D7 ^# l% W7 ~4 O: I
8 e0 ?4 R. x2 k. J

& r2 u' G1 y6 ^3 _# k" {: lintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b) 4 a& V: ~# o( e3 D7 X" Y
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
( ]4 l# s1 U" N$ Mtherefore:8 J- b. _2 x/ S6 }9 U1 F
3 t3 y: _+ d5 ^2 p
{(a+bx)/K} dY(x)/dx=Y(x)
- p7 R6 a" T5 K, o( G( C; a) k1 z0 F% R  y1 f; `
from here, we can get:
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3 _4 N2 b4 ?  AdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx). S" Y) X: J# P! R

1 i6 G" C& X7 Qso that:   ln Y(x) =( K/b) ln(a+bx)
, w; O& \6 D. O% B* i3 P( F3 c$ G6 r% l/ W$ u* U' [0 Q! |# \
this means:  Y(x) = (a+bx)^(K/b)
/ @4 D4 ^5 c3 r, W9 Xby using early transform, we can have:
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-(k+b)C(x)+s = (a+bx)^(k/b+1)' j0 b- r$ M$ ?% W( A

. E1 d; C! d$ H' C# c  _finally:- }* T: U' ~' e; x
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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:
& i! u, Q' ]) i3 X9 M% H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!

( M$ T2 J  C/ d, U  F. \/ }
; J& O5 a, R0 K0 N0 F7 T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。% `' H& j# m6 s% A3 ]1 S
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。
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发表于 2005-4-6 18:34 | 显示全部楼层
9楼的答案有点问题吧。
3 s* K" s- V, }; [2 W您说:
7 \- [$ T, i4 f3 ^; _; X! u4 t& H/ y4 o. s3 I
d{(a+bx)*C(x)}/dx =-k C(x) + s! [' D5 g, T1 M; u( c
so:
" q3 u5 R5 [+ F3 y: p5 I/ v7 gbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s3 Q5 M* B' Q0 Q7 C- e" l& @4 I

$ B( Q' E+ Q7 l: c, `* v也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
; t2 ~8 i: ~5 Q: y. B1 b+ H但这是不正确的
, f) ~$ A5 U, N, E( P  ~# `d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
0 V) W& ]4 o: @. X; P% \并不是   (a+bx)*C(x)   的导数除以   x   的导数。; G& Q  ~$ Z9 L* x8 j8 q( }) ?3 r
比如说   dx/dy = (x'y-xy')/y2
) u* r" P; @7 n2 C# ~x' 意思是x的导数,y'意思是y的导数。分母是y的平方。% h0 Z/ S3 T0 s+ v  }+ s
因此:
# l& L# j, u  k2 M9 ~5 z
3 Y7 f3 c/ a4 _7 u6 A- Td{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2/ i1 `4 h! R6 c. j5 q
最右边是x的平方。
! {% F: L" \, [3 N" C2 z% p  H9 e
( i9 [( C0 y3 M- [所以您解的有问题。
2 d2 D' N' @0 C$ K  M  {7 n. U$ s9 ~. }9 v" @
[ 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:
9 u8 |/ s$ Z) m9楼的答案有点问题吧。0 |( ~* ~1 i+ [7 X5 B
您说:9 d, G+ R  m; ]% [, D
4 V. G3 N; e# {4 m+ `$ t
d{(a+bx)*C(x)}/dx =-k C(x) + s
3 y7 b0 R. J. {/ ?3 M- Z! Pso:
, ^8 Q  f3 P! |$ K! z( b& u+ YbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s! S) F3 n$ _8 r" Z* I

8 D2 ^. ^* S' M& ~也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
/ P5 y3 d: ]' D* d/ x/ y3 n但这是不正确的
9 ]# X$ r7 N/ M7 b( F9 D" ud{(a+bx)* ...
& a1 G% l9 v+ @" x: z4 p) f

1 W6 f7 w; H& y. J& S$ W# c# qPlease note here:
6 ^" r4 r$ j9 w+ z: s1 k' P1 S% T+ m5 ~8 R3 T* ^/ A+ r0 |
d{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
% t8 k* [7 k! V/ |  \/ ait 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
$ }) K% b( N: Z- F1 g8 M( Y( dd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
$ A4 [7 [& _# {4 O8 N1 a3 p并不是   (a+bx)*C(x)   的导数除以   x   的导数。
4 T, z. O9 X7 A  g9 y& x! n: E7 S; o
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: i+ N, {: Y9 R1 W
....- J" P6 d$ l; y( A. L
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

3 u' d+ s  l4 g0 a5 \# u3 e3 x& H* c, w  D1 p4 Z8 ?, g
这样算混淆了微分和导数的概念。
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$ S. H& [5 `$ V+ [' x8 _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3 T# ^0 |* Z: l7 E
9 O4 [/ X# p  f* H1 a, b/ {
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)
" h% S: s: J, q7 R: ]% Z* `' C
[ 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 X4 c/ J2 z. e" p这样算混淆了微分和导数的概念。* m$ N% V5 k. G8 Z- L
( c- O$ a) L1 G# Q
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
- O8 _% W0 x* ?/ k' v- ]* C8 \" F+ v9 B" T* |
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...
' }. y% ^6 @) z

( Y7 S% H, J. ^. R8 Y2 r: n) B8 [这样算混淆了微分和导数的概念。
) V- d2 W6 C2 S! N7 g: l( N% N9 B* O4 c3 X. Z3 }9 T& j# \
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4 U, o6 P! E4 |0 q$ F: c
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Ah, you made a mistake: it should be:5 C  h/ g4 J/ K7 l6 g
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d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx

; F2 H! u+ B1 Y, g3 W7 ?& F# o; e( i& K! C) N7 y+ d
* V9 I/ D; ?4 X- i
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)
6 }& i. D& ^7 n
; N* d4 M+ `- e- |2 Mfor h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:
, F; Q2 {% g5 h" ~; `8 j, m
! Z" I; Q, g: }& _7 [h(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:
; {! }. P9 u1 y) n9 v1 f0 |+ Y* U这样算混淆了微分和导数的概念。
5 x+ `! E* M. g" a4 O
, R, u7 ]+ q' T8 ?* H. W- r5 n3 Ld(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
3 F' H- l" Y5 U. ^: g5 o; L
8 z6 [" p, z3 E) LAh, you made a mistake: it should be:& K7 o+ C0 ^. _( d

1 }# }% V4 W0 T: ~& r5 h6 i- t' [d(f(x)g(x)) ...
" ]. ~- z- T9 h+ g

, B6 I; z+ h$ n4 e0 V1 c: C6 ]Yes, there should not be " ' " after "f" and "g", it is a typo. :D4 L6 W7 E8 b3 @% ^$ |1 g- ^# |8 a9 q
2 d2 ~8 H5 `5 n' q2 l# [0 `
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 20:51:2 w, g, W) y7 Y8 C4 i9 Z: W/ J
Yes, there should not be " ' " after "f" and "g", it is a typo. :D# u/ |7 q/ m% R& M; h4 P1 s+ W
. c$ B9 H! D3 H
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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9 P! I' o& X" k1 T: h2 _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:
! }- M* u; \/ ]. ~+ D$ w$ I# S/ |* L: t. F2 ^9 l3 G
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。
) L2 Z% r. b' z( s& N% _5 P. t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 ...
* q) k7 o4 H3 I0 ^
7 C6 s- }' d2 h9 N. b
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:
7 Y8 u- B# c2 l  X4 Z' ]7 ^+ 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?
. w9 `/ D9 d; }
- g( [% _; z0 K- z# K/ p5 q/ N" 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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