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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!
/ F% U, K; n) L7 G9 t* K! a3 }; P3 ^) l
d [(a+bx) ] /dx = -kc +s / N# M, P' e; X" k- y5 K
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
1 K0 S- h2 J4 D! d) \& Z& n4 X$ f( J7 c7 t- E3 G4 B
多谢了!
7 P6 f  ?- {' t4 \% ~5 f
! K0 J5 r4 `- ?3 ^8 k' c[ 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:
: X# c+ a2 J- w0 V1 w" h+ J% O4 Q* e请教大家一道微分题,多谢!
; P2 c- ?0 B# z# Q5 |2 C9 Y
! j& s7 _4 \/ ?d [(a+bx) ] /dx = -kc +s
% U3 Q8 w1 J2 c* [1 c0 |: {where: only x and c are unknown, others are all known,  requiire c = ?0 n3 \$ u7 z! K2 W% s. w9 d
6 B! b) O8 w$ H( Y/ t
多谢了!
# G9 U) `3 h7 S9 }, y/ [

  c4 T7 B* r6 ^$ Rit is too easy:# o5 Q% b0 t1 L- s' ^" ~

0 ?) H5 q$ s' i5 E& ld(a+bx)/dx = b
, C5 I/ e2 N- n- Z( L: e5 b( x% Y3 S' c
so& Q6 \* T3 \1 h

# m  S! K1 a  d5 Jb=-kc+s
; W+ Q8 M1 a5 E( A, n# Y
8 Y( b' X) y" a6 Cfinally:  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:
( F6 C, T, l$ a; j3 [it is too easy:
! J+ m6 }$ l+ {; h3 {4 O1 T1 u3 @
d(a+bx)/dx = b
8 P8 a, s1 O+ g& R! }$ b4 r) S3 _* }
so) K5 Z0 {' y- K  R% p

6 C2 F4 z* k$ }7 M1 zb=-kc+s) b& _: }* T- `# n9 s

/ }2 @  G6 a: E3 Xfinally:  c= (s-b)/k

3 m7 y) g& F% g$ N3 ]3 J. A; ^7 [; @) 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:
# n% I* i* l& y. ]4 o/ [: T* Z请教大家一道微分题,多谢!
; i% k6 m* m) F# T) d0 v/ Q% ^2 h1 V* d
d [(a+bx) ] /dx = -kc +s
6 F3 i$ y; b! [/ P# S1 t, wwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
- |, H- h1 ]9 B" a5 h/ t, a1 c, ~0 v/ j  z
多谢了!
+ D0 \  V: Z4 p, f  {
" H4 T$ V# p6 A" C# w, S4 j[ Last edited by 醉酒 ...
4 E8 a/ V: w2 x  I; q9 J

. ^4 X5 W8 j1 K7 Y6 U  K/ C3 l! osorry, did you post another question? your statement is still not clear.
- Y; o  @7 ^* I# A' G/ }" S: U( G
, N- S- i& `8 Y, S7 ua, b, s, k are  constants? c means c(x) ?
1 H# h) h2 |' T4 r/ K1 c  k. \) H! K) w( V# a" T  A
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:. v/ G9 y" x/ D. G" K* L: V
sorry, did you post another question? your statement is still not clear.& X# g; }- C$ o1 @1 Z

/ U0 h: b7 J1 x8 Fa, b, s, k are  constants? c means c(x) ?, ?2 j& H9 h9 w( B  n& e
& p8 t, }/ S( d) P: n, u
please tell me.
2 a2 {" A7 M% ~3 I' ?
对不起,写错了,忘了变量c,应该是' G8 J7 F6 g' |
d [(a+bx) c] /dx = -kc +s + H0 v7 K) Y% w5 v0 b

  Q* A7 w. f. R0 O0 d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:
+ y# W3 A# o( N对不起,写错了,忘了变量c,应该是4 ]7 u8 x( I% M3 [
d [(a+bx) c] /dx = -kc +s
  T% B$ ?" [8 ?. ^/ ?* N& q( ?
6 T* R. N) `/ d0 g; L多谢
" h6 o( D) d! {- h4 B8 H1 P

8 g1 n( m/ y" L/ O: i  o) f9 tdo you mean it is:' {2 a1 h( z( `7 t
' a) Q) c9 d  R2 P+ ~( {5 ]
d { (a+bx) C(x)}/dx = -k C(x) + s( q! b- @; u, T6 E7 o
0 o7 a+ @# G- A! ^( z. a2 u
where a, b, s are constants, you want to know what is C(x) ?  u4 L; H1 C$ f/ C

: @6 e8 G! s1 v+ mif 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:
1 g8 p+ x2 ?+ |' a4 V: t( Xdo you mean it is:% v4 t! ~" O5 a4 S6 g, }1 O

' G  ]- f. m8 \% vd { (a+bx) C(x)}/dx = -k C(x) + s; o( @  [! h: ]. P
0 S, O! [. }7 k2 u( L9 q/ l* [8 h
where a, b, s are constants, you want to know what is C(x) ?  ~$ A$ H" z! F  A( x% M9 @$ _' w3 Y
9 I: u9 \$ W$ v$ p7 @
if so, this needs to solve the differential equation. please comfirm it fi ...
6 K' `% S1 H9 c8 R, [
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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
老杨团队,追求完美;客户至上,服务到位!
this answer is the good one.' r9 k' y/ `+ Q4 W1 S! h9 w

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+ g* B; [- d! y* jprocedure:
# n! \, @$ |% ]  L( @7 I4 b# a6 N* J4 a- w  t( T4 l0 B/ b& ~* M
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
' G( L7 U( I2 f6 E5 Yso:5 M( i0 ]/ m5 ~4 v
, m9 t7 R- p' D
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s+ H- X5 C5 Z3 Z) `% R/ l
i.e.
( ]; i% A0 e" i9 }/ M/ U. H$ h9 n* b
# E  s- h. N9 n8 y% U5 f" h  ^, S(a+bx) dC(x)/dx  = -(k+b)C(x) +s- o1 K8 m" s3 e$ a
+ ?4 \. r: L' n0 ^
8 o" ~# D5 {  ]3 }, s! u8 G
introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
6 H2 r4 h& Y: b, j. |which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
3 D1 `+ p" Z0 S* p+ Utherefore:. l% N1 y9 {8 L3 R1 a* }
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{(a+bx)/K} dY(x)/dx=Y(x)
' Y* Z% z, [, Z7 r3 q3 v- ]! i' ~$ P3 N' l2 x7 x; A. p4 ^
from here, we can get:7 I+ e8 \0 n5 a# k. F- Y: z
2 y; a3 H9 ^7 Q" K
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)" s+ u0 `1 D% i, `$ U8 G
. J3 `1 h* K* ^( A
so that:   ln Y(x) =( K/b) ln(a+bx)! C* h3 X6 p+ C% {* R. K4 V
+ _. \* X: c9 E* V% ~. c
this means:  Y(x) = (a+bx)^(K/b)1 T% W5 f7 U1 j( h# t
by using early transform, we can have:
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-(k+b)C(x)+s = (a+bx)^(k/b+1)
# q1 O6 B5 Y5 {5 x! z/ H" n% @1 y! e- y7 i! f# l! g, Q: w& z  v
finally:6 i  Y  g2 u1 A( J3 \, J$ `: {9 H* }

  r$ r" t/ s$ Z4 H4 I* H' ~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:
; n) g8 M, \5 i( z2 P$ I8 v: c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 R+ \$ p. E/ A6 k6 c  |% x6 r9 g" A
$ U+ s# b5 I* z2 H3 d' y
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。+ p: _8 A# |% e: B& c
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。
9 G, G9 }3 P4 N5 L7 z您说:, t  u8 v  Q3 [
2 _+ q  `8 k8 E* j$ t. O* [
d{(a+bx)*C(x)}/dx =-k C(x) + s& E) d* ^: z" ?) Y
so:+ \+ x# X7 P  j( J8 L. d3 C  ^
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s* _5 U: \+ B% f

. a( `& y9 n5 ^  Z9 V$ D' }7 A! B# I/ V. r也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
& H% Z* ?, f. K$ K, K- }但这是不正确的. Y% R1 ^5 l, U2 w
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数  r. o7 U  r6 H
并不是   (a+bx)*C(x)   的导数除以   x   的导数。
, P/ C5 M* _6 ]$ b3 w# A( j$ X比如说   dx/dy = (x'y-xy')/y2; b0 n! T( X+ X% H( C7 [4 S; @
x' 意思是x的导数,y'意思是y的导数。分母是y的平方。4 F! O7 ~  r# h9 ^9 l
因此:
+ d9 P. s, o- Y8 W* N( E' p( j" a
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
4 L. L( b' v  ^最右边是x的平方。
* b8 r9 n. `  h' b1 p9 x6 F# B9 U6 [% ?7 }
所以您解的有问题。1 [  L* E& G; l
& e+ p( j# ]. z6 w
[ 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:& U( o! l( f+ ]* [4 L& }, S. b/ h+ O
9楼的答案有点问题吧。8 j' S0 B  M) K+ u" u
您说:0 ^3 _+ r& D8 {4 x

9 F* B; N3 C! u) }9 Y/ E! K/ Md{(a+bx)*C(x)}/dx =-k C(x) + s
! Y% I8 {5 k, u7 T0 b' O, u- L. zso:; ~% O: ^" ?1 o0 w0 ~4 O' e
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s4 A! M$ m5 E$ N# e
; e8 C" a- ~: X8 V: u( f! l# Q8 T
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx3 _0 t$ c7 ]7 z
但这是不正确的
: {" ]1 U% q5 u6 C/ S; Hd{(a+bx)* ...
" t! N6 ]( O  z% q6 @) a1 ]$ V

7 \: R9 n" g: o3 q# G- U' x  hPlease note here:$ u+ N' o3 l1 V5 D' @3 U' v
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d{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)  S# `+ {" y& a) k5 I8 c* a( N# x
it is a multiple, not division. Right ?
大型搬家
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
) Z3 _* D, T) ~: Y. H* Vd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数( G8 t" T; x4 C( R0 u" u( L
并不是   (a+bx)*C(x)   的导数除以   x   的导数。
) C, t1 U4 \5 Y% [0 t5 t  _) B! Y0 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" ?' Z% y9 A, {, ]) |
....
3 k5 w" P0 w& X9 t7 r$ i" H% Zd{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

  {$ i) I7 T: A* C. I" o* U. ^+ Q8 \7 ^7 k  M
这样算混淆了微分和导数的概念。2 ~! R  Y- ?$ S; ?
* _  a% Q4 v( B3 W0 t, ^) I4 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
( ]! p: m6 B% r8 h6 j" T* g( m  D8 {9 K0 K4 \9 `* j% X
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: V8 H9 v1 c1 l

5 V' o4 V$ E) |8 g1 D1 |[ 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:
2 q3 g" m' U2 j) _1 I这样算混淆了微分和导数的概念。4 @2 a$ C( ]  Y

' k9 r  L7 y: Q( O( j4 A' i% od(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 k/ `9 o% V9 t3 j, w  A

# Q/ U* c3 I% T/ R4 X' fIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

. |  R8 T* s% g$ ~3 }+ s
3 ]  e  w* A+ a: D! R( b这样算混淆了微分和导数的概念。
3 a' r5 k  T8 v$ x2 X  W! t9 B4 e# t: S: `/ L3 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
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1 L! x" t$ E  T- l' nAh, you made a mistake: it should be:
$ T" n# i5 X" R! d. p6 o
9 Y1 d! x  s4 T  t( O8 k0 ^d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
- b& R% E" u+ Z* w4 g7 [
  T! L# o1 I7 F5 V* n8 b

( C0 N4 d9 }3 s8 e4 k9 P5 _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 w' x5 C) Y- \. H! z# |* t; h" o8 v0 M; y; x& R, ]
for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:! Z3 N' C% z  n" \3 x3 L
, k3 w0 [5 e1 h$ t* X
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:9 n" T/ L# K* @0 @4 S. w1 j$ F
这样算混淆了微分和导数的概念。, z: H, K# {) Z# e

1 e( Z* n( e9 f( vd(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
& g% x+ {7 n8 n) g- ^& P& Y/ [! M7 A  C
Ah, you made a mistake: it should be:% b8 b# V4 M8 F) I+ A$ l4 V

6 Z& g& l0 d; q% r, ed(f(x)g(x)) ...

2 F" Q4 T0 M$ h6 T7 c! |0 S, |- V, ?! J1 t6 a( }
Yes, there should not be " ' " after "f" and "g", it is a typo. :D" S  n# _0 G. V: X" y
1 L% B: @* Y2 u( [
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:/ n# i+ |1 n  e6 k
Yes, there should not be " ' " after "f" and "g", it is a typo. :D- c& Z- ^' k. |5 V, I( Z
$ ], x" z% S1 I3 n
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

# @0 @# D/ U( O  ~8 D
! Y3 M+ J: P7 V( G; d# rThanks, 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:
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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。
; c* C) _5 w/ J. F0 `9 B! y, K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 ...

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( y% u8 E! v) q4 ~& }( h6 m$ T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?
理袁律师事务所
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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:+ V0 Q6 z0 B5 M2 r  {/ C3 s$ `
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?

1 M5 q2 R# z; l* r% c) e# e
/ H6 d7 R4 h6 {; i5 u% M) Q+ Q* 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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