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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!* t3 g( d6 J/ s: [" o* H
' l6 V, E4 x9 O3 A# W
d [(a+bx) ] /dx = -kc +s
6 N* [: [( i, S/ j/ X. Q; ]& `1 v6 Lwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
, `* {* [$ }: W7 P; i/ A0 o" J4 a: b
多谢了!+ O; h3 E& V6 J

! k& R1 i5 ~& G3 x/ e- Q[ 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:
* P- [9 R$ F6 }- J6 t/ g请教大家一道微分题,多谢!* o: b; Z, z: {$ t( i& |. a
7 H5 ]" a8 D, v: A5 \
d [(a+bx) ] /dx = -kc +s
9 q9 a7 k: q4 s+ t( |+ ywhere: only x and c are unknown, others are all known,  requiire c = ?
3 O# V+ }: ~' O; i8 A2 w- S  ?% m3 _( f
多谢了!
2 d3 s  s7 h: r1 y  A4 m9 u
- }6 N  R$ v  `7 ~4 t2 A4 R* M* s
it is too easy:6 t( v1 Y$ f/ w5 a: R, M

! W2 a9 G+ i- m6 E' rd(a+bx)/dx = b' T: P- R/ I/ C& a2 u/ z) E' }
* Q! i; r( n* l& `5 z1 k, ]
so1 M: F3 V% j. z9 H- j  Z) D

0 F1 b0 G3 K' ^+ t7 a, ~! lb=-kc+s1 }9 R7 D+ L; G. E

7 n: m" D, R; H3 R' hfinally:  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:
: x# x+ \" S3 C1 Oit is too easy:
2 J( F, _: B: @
& S7 R! M! o9 \0 p3 E. t5 nd(a+bx)/dx = b
' ?! Z3 w" U7 X# D( A1 a2 N- y, |% C
% y- ^3 C; t) l& @5 R7 ?so$ [/ f4 J$ {  I. w/ j/ A
- ^7 Z, A* Z& x' |9 s0 w: x
b=-kc+s* U8 R+ `9 E! [( e
- P6 K* V9 W5 `) L
finally:  c= (s-b)/k
* z% ^2 D; }, \$ y, l/ C$ f/ j7 }
4 s( P6 n+ D: `' G. Q7 x4 }
the right side cannnot contain c, but b contains c, so it is still not solved.
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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:
) A# G% i8 C3 c# f3 d% E请教大家一道微分题,多谢!% A' j& P0 e, V+ B& y4 W
! r+ P+ J* n' f. K
d [(a+bx) ] /dx = -kc +s
- b4 L$ `8 z. Pwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
6 T9 R. z! i  z* v4 ^' x  V/ T+ Y2 v) L4 w2 @& I2 V& Q& X" t. _" f4 k6 ~+ ^: q
多谢了!
+ }/ X3 U4 z# u) u! z' |  }8 t5 P' t5 I) X8 s
[ Last edited by 醉酒 ...
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sorry, did you post another question? your statement is still not clear.) E4 {/ f/ x- Y$ o$ A( [7 {
9 }. L& g* I( v6 x. J# b  O$ K/ ?
a, b, s, k are  constants? c means c(x) ?
, X. s& `$ i4 Y  f9 W6 k( @1 A
( [% z2 z  ?& P$ Jplease 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:2 n8 s$ N5 w  o" l5 \+ C" R$ n6 B3 P
sorry, did you post another question? your statement is still not clear.
, ~( k# e& r. _: x0 {: U/ f/ ^+ j& v) C
a, b, s, k are  constants? c means c(x) ?' N# R/ p8 l# H) j6 Y8 ]; b
% }) A/ z0 w* g7 J  O
please tell me.
3 [( k& ]! M) O9 I0 ~: N" [0 ~
对不起,写错了,忘了变量c,应该是
2 s/ n% F/ s, B0 U1 \, n, Sd [(a+bx) c] /dx = -kc +s - f- S, ]' z; E( M

( E- k1 [: c2 R5 x, H, M& f4 K多谢
理袁律师事务所
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:
7 x6 O. J5 _! _, e# F( b; u对不起,写错了,忘了变量c,应该是
; t0 J2 k- J* V& B% _- Qd [(a+bx) c] /dx = -kc +s
4 f# N( c( H- P4 n5 D3 j. D! t+ U' H4 W
多谢

; W! D+ \  y$ T& w- j" Z7 w# s9 W5 s1 T" }
do you mean it is:  v# q7 T8 k; {( z8 P

2 V1 m+ j! m9 {d { (a+bx) C(x)}/dx = -k C(x) + s
* m5 f! f, z. k& S9 g6 b; h& J. j4 W( x- `0 M
where a, b, s are constants, you want to know what is C(x) ?5 ~* A4 M& b" a& R2 T4 J+ D
; U9 `6 u6 J! |$ n
if 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:7 s/ h% ^* R( [. V4 M
do you mean it is:
3 ~5 z- p" V% Y, H" k* Z9 H0 V" M+ @' l1 \, r. l
d { (a+bx) C(x)}/dx = -k C(x) + s
/ M# a  w0 x7 r2 u/ s$ K" {+ N$ `; a$ V% q: ]7 y2 ]
where a, b, s are constants, you want to know what is C(x) ?
  t' S% H6 X1 o: d" [
( {6 A6 R- E) h/ w1 qif so, this needs to solve the differential equation. please comfirm it fi ...

- d0 I/ @2 h: a! }- a0 }5 A, l, j, F/ G/ l* ?& t" ~: M
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:
5 c( i3 T3 z/ a7 V& D9 D5 I& y$ S) G( j- m
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.
, u! o8 {/ M* T! b8 o: u5 H3 r' p* Y/ v) G
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procedure:
8 P$ ^$ S3 [& V, A
, h1 M8 S2 {0 e4 @From:  d{(a+bx)*C(x)}/dx =-k C(x) + s: M% n& |. R# z
so:- o& j* }4 S8 C

$ o% n: Y- [6 GbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
9 c. W; F6 J. v$ Ji.e.: b4 ]/ a$ F( s! I

& A9 Q! u$ N7 _7 ^1 k(a+bx) dC(x)/dx  = -(k+b)C(x) +s9 f& k. D* H) _' ^( V; v5 D3 C

" v- T+ L7 J& a+ V# o+ x
/ h, l  Q9 l7 ~% o' ?* G( lintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b) ( |! M  j: V  H2 C' t, D8 p/ y
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx7 X1 n% G1 b, _7 ~" F3 E
therefore:
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* h7 C% x1 G. i/ W0 |  i* F{(a+bx)/K} dY(x)/dx=Y(x)
. w$ h4 W9 \! G, ~4 G+ d1 D) r! B" `9 F" x% I0 R
from here, we can get:
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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)1 G. ?1 Z! n6 R' ^! t' d
4 M  w4 b* s: ?0 b) ~8 ]& @1 B; W
so that:   ln Y(x) =( K/b) ln(a+bx)
/ [6 |" W/ P& m6 J0 Z0 @3 ~
' `) X! c8 H" }& B0 jthis means:  Y(x) = (a+bx)^(K/b)) S8 U9 C' y; Y  E/ N1 W
by using early transform, we can have:, B9 E2 S8 @7 C( ^$ u6 d8 A

/ x, H2 N2 J& A1 K- C-(k+b)C(x)+s = (a+bx)^(k/b+1)1 y/ N+ f  v; }2 L
+ q4 F1 n6 o3 t. _8 @6 y. r
finally:
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5 U' k. V6 ?( W* NC(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 Y4 `$ T! ?; L/ W
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!
2 u' ?0 K! x) u" L

- m0 L# [7 O+ K! n2 G6 w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。
  j) H2 w% A! B; f- a/ u2 d% e! [$ i6 G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。
% ]% ~6 |/ `# x您说:
6 I4 L) J0 g2 H) ^; g0 X9 D1 \5 p7 |; V7 t6 c" h; e) V' f
d{(a+bx)*C(x)}/dx =-k C(x) + s* W! L4 \' V. ^, b/ H& T# K. N, W; b
so:
( F8 z) Y- X. m' m* KbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
9 W7 h+ l& n  v/ f; V7 ?0 d
3 r& d' |! t* f% w% l4 O2 f也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx' N: X( N. G: g5 \
但这是不正确的
. o3 S! K- J0 u+ 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
! t, W4 N" ~; S- j, |6 S( O+ q并不是   (a+bx)*C(x)   的导数除以   x   的导数。2 I, t- g9 u, }9 U
比如说   dx/dy = (x'y-xy')/y2
+ a) x' l! V9 ~. M/ b5 }! Lx' 意思是x的导数,y'意思是y的导数。分母是y的平方。# Z% D9 r- y3 y6 d6 Z: E
因此:- r, j# h* j, E/ r* ]! r
8 d+ C5 @4 D" Y2 }
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2- w0 \! Z' p& ]1 Y+ {1 B  ?
最右边是x的平方。& L9 C/ _7 Q: C8 Q
- r  P# `  r# g: E4 e6 M" B
所以您解的有问题。
6 L% A: d$ H! D6 J
* O- f9 w* j" P[ 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:" B3 K' r. W+ _% l, n
9楼的答案有点问题吧。/ n2 s8 f3 x  Y5 t3 N! m7 ~/ [8 s
您说:' L1 n  `. R" W5 f1 b; ~
+ G8 ^: y7 Z' w
d{(a+bx)*C(x)}/dx =-k C(x) + s
1 g; E6 Y( u$ C) cso:
5 Z* A1 n9 c5 K3 ~$ mbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
' ^/ ^4 j( x- N. Q. c  f. S& j; R1 l: s
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx" x* R& e' {+ w) q  v
但这是不正确的! w: q+ }5 M# P$ D. Y
d{(a+bx)* ...

5 |4 ]+ i. w, s. s! m
8 l4 D% i+ T0 W" m' z( SPlease note here:
0 }" R  W, c/ W6 o- {3 ]- C; ~/ e3 a: s+ l# O
d{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
& ~# D7 y% J, a" ?# r( ?it is a multiple, not division. Right ?
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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
  e1 {4 C) r3 j: c) M& N: V* Bd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
1 p- n9 V6 ~/ v$ ?7 I并不是   (a+bx)*C(x)   的导数除以   x   的导数。8 m2 a( X, n0 F# ?( R0 h& r8 O  p7 }2 t
* ^; b3 e4 W' n0 w
no "x" in that position.
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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
7 [2 m5 X; x, q! y....9 E& o: c* p0 b6 n$ j2 M" L( s' ]
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
5 S3 S0 w& T+ p7 E3 U  S

7 A9 {- u/ F8 H/ w; B2 A( W这样算混淆了微分和导数的概念。& c, P$ j* l; B, f+ A+ {: `

$ a/ }* q7 {3 Ud(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; ]: ]: ^; R2 ?( ^  N5 T' r

8 V7 V6 |% F! t- {; ?6 f  vIf 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)" q5 i: ~' c% I9 Q* j

& K) X% Q9 r9 g; w( ?$ ^+ I! c1 |[ Last edited by gnome-kde on 2005-4-6 at 08:31 PM ]
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发表于 2005-4-6 19:45 | 显示全部楼层
Originally posted by gnome-kde at 2005-4-6 20:30:4 c  L1 l) B1 _- F. i5 D
这样算混淆了微分和导数的概念。
8 X6 E; ^6 x% {0 n1 C7 g6 U# ^& F4 b  p2 S9 m  l9 O
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 A) s" j$ z4 {! v: P
% E8 e8 C: B2 ^
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

' L+ J& q1 `8 @, j6 t& [, q
* l, n1 o0 M( k这样算混淆了微分和导数的概念。
8 Z6 Z1 J  |  d. D# i, w
! d; Q$ e, v+ Y9 P4 M0 }+ Md(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9 ^6 h7 m; J7 _2 P( T

+ e2 |  ]$ G4 ?  ^: h0 `Ah, you made a mistake: it should be:" [, }' p  q$ p2 F+ U5 ?

- _- T# K0 {7 N5 L* v' N8 \3 c. Td(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx

0 j$ O$ ~9 r; W( g  B3 f4 u% V' P/ c+ A0 K
6 {2 G! s3 [, ?2 h" 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)
4 R2 W" C; W/ C. B  r, l1 u9 |  F% L" D$ Q3 z
for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:
$ U3 `6 ?+ _7 [: x5 z9 `* H
* ~+ P+ ?" v3 ]- V0 |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:
7 P$ A9 @! l% W0 E' e这样算混淆了微分和导数的概念。5 q: _8 `4 ~9 Y% T2 n
. m4 e. x# u. C7 X) [
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1 i3 v* X( |, \0 f
) s  g8 i0 b% y, m
Ah, you made a mistake: it should be:7 y# O1 H$ t0 _* F6 @5 O# }+ q
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d(f(x)g(x)) ...
6 m' @6 f3 h& u0 H% C( [
3 A/ B( W  T# h7 ~$ n3 _
Yes, there should not be " ' " after "f" and "g", it is a typo. :D
* o8 K" ^# [; X
8 r% c6 M% q( bBut 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 20:51:1 x, U. X: T! ?9 t4 D* l
Yes, there should not be " ' " after "f" and "g", it is a typo. :D1 L& B6 O& V! G6 j2 [  p

& T+ K7 A0 ]" {But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

1 P! {+ D; U0 E; l9 g! z# ~- |+ t# K9 c/ c! `0 E  _4 x4 n
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:
) `' H/ U) I8 ^7 H5 z
' R6 ?5 Z; B+ P; Z( M7 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。
1 A/ `, |* B5 D; E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 ...
: _1 X+ u; G3 W, g# E- h

) t% R& u6 u" m& `) v: `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:
* r0 a' `! m: F! b1 j" U. I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?

6 G, m& S4 D2 Q6 h' {# [3 t, \7 o5 [) Q5 G
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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