数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！
2 f8 r2 c8 ?0 r
d [(a+bx) ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
+ I. T9 o, t1 }" L: W/ q0 }
多谢了！
6 V# _9 G" a! `8 p, U  L& Q
( m$ Q  g  ?9 k6 |! l" d[ Last edited by 醉酒当歌 on 2005-4-3 at 07:56 PM ]

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-3 20:41:
请教大家一道微分题，多谢！
/ ?* W" T3 p( b7 C
: h7 w+ |$ s8 r7 h  ad [(a+bx) ] /dx = -kc +s
& q6 o4 n: ~2 uwhere: only x and c are unknown, others are all known,  requiire c = ?
; n$ r- M; v" y8 Y2 E
多谢了！

6 ?! N4 \( X; l: j3 L% t
2 h* B0 a" |# S+ git is too easy:

( k3 |6 k0 e7 \% Y' z9 f) Sd(a+bx)/dx = b

& a# m/ |/ W) I8 Y% N7 Yso
  f* S4 `! T7 t/ s1 T
/ m" G& B! K8 ~. Jb=-kc+s

finally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
5 o  L9 i1 {6 eit is too easy:
- j/ _' T( }. w2 s9 N1 O6 a* d2 V
d(a+bx)/dx = b

8 K$ j, @6 f: h  f) Cso
; F& ~6 B3 C% X
8 D; ]% ~' R! \! F, E( \+ _  Pb=-kc+s
; _" \# R2 H/ \$ y. V
/ r/ r5 v' P7 e0 Z: x- _finally:  c= (s-b)/k

  i3 \; }' n/ `4 U3 ~" E
the right side cannnot contain c, but b contains c, so it is still not solved.

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-3 20:41:
请教大家一道微分题，多谢！

; p# d4 t- h! d) q, `0 S! |d [(a+bx) ] /dx = -kc +s
; S7 ?7 T3 P0 z0 U( S: r, i# `6 qwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！

% V$ a# r: x: M( O, _[ Last edited by 醉酒 ...

sorry, did you post another question? your statement is still not clear.
8 y+ c: n) T& r9 r0 t0 I5 f$ n8 Y
a, b, s, k are  constants? c means c(x) ?
5 M2 _) V9 M5 b, Z5 ?% m
please tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
sorry, did you post another question? your statement is still not clear.
) |2 s, c! A: H
; C3 r6 b" Z# pa, b, s, k are  constants? c means c(x) ?

please tell me.

9 T" j9 i9 l6 w2 B) [- k对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s
7 O, \5 r) g+ l( R" \7 ]
多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
  }6 G+ K9 Q( E1 N  h3 q! s. T对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s

多谢

' x" ?$ B, X2 e1 {- Tdo you mean it is:

d { (a+bx) C(x)}/dx = -k C(x) + s

where a, b, s are constants, you want to know what is C(x) ?

3 d. \+ h4 ?4 o- O/ V, V1 Jif so, this needs to solve the differential equation. please comfirm it first.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-4 06:36 PM:
do you mean it is:

d { (a+bx) C(x)}/dx = -k C(x) + s

1 T9 N  {2 T8 V7 y) I5 Nwhere a, b, s are constants, you want to know what is C(x) ?

if so, this needs to solve the differential equation. please comfirm it fi ...

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。

十年移民路_

answer:

C(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.
( Y& g) x8 v$ g8 k0 F5 s
6 L2 F. s0 W1 x0 }
procedure:
& }1 P+ p) A7 {5 Y
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:

bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
7 m- b$ {7 F! r" ^" S2 Y9 E, n1 ~i.e.

) V9 R. U! L8 j5 B9 K! @(a+bx) dC(x)/dx  = -(k+b)C(x) +s

" T' K1 [3 f  f% b/ I* w3 M9 Z
, K- `, ?3 p' Yintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
therefore:

0 D+ v5 T9 X  o8 _, d2 C{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:
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8 s6 }' D8 `( M* m1 O: PdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)

/ P8 D+ r' q" m1 Q3 N% oso that:   ln Y(x) =( K/b) ln(a+bx)
$ u3 ?* f- H8 E
this means:  Y(x) = (a+bx)^(K/b)
7 x6 p0 X4 C$ L6 ]/ _2 rby using early transform, we can have:
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-(k+b)C(x)+s = (a+bx)^(k/b+1)
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( J( x% E! z. ~, B8 Hfinally:
8 i9 j- J) p$ C9 x
5 `" H" X1 i; U  sC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)

醉酒当歌

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！

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-5 14:33:
: A) G. v( W( X; C+ q1 [: B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！

+ L  {" x1 C% A0 ~0 I' H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。
- G% y) a. p. P# ^' U! r$ u. k9 d7 \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。

龙舞九天

9楼的答案有点问题吧。
您说：
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d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
但这是不正确的
/ m! I6 w2 G- u6 I2 {9 fd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
并不是   (a+bx)*C(x)   的导数除以   x   的导数。
比如说   dx/dy = (x'y-xy')/y2
x' 意思是x的导数，y'意思是y的导数。分母是y的平方。
因此：

d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
最右边是x的平方。

% B3 `; _" _) u" C: {+ C所以您解的有问题。

( r% A! K+ m! f, F2 I[ Last edited by 龙舞九天 on 2005-4-6 at 07:35 PM ]

十年移民路_

Originally posted by 龙舞九天 at 2005-4-6 19:34:
9楼的答案有点问题吧。
您说：
4 R. z7 l4 W' t
d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
  u* l; K  S7 H( }# y0 w4 mbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

$ U' E3 b" K3 L也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
- d0 x, o8 G9 n- A但这是不正确的
! z& d$ h) m3 F3 Qd{(a+bx)* ...

8 X( D9 }! p8 l. Z
Please note here:
& \$ T8 h$ U6 z; y. `) n- X
; t7 y2 g& J  G2 p. a9 Fd{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
it is a multiple, not division. Right ?

十年移民路_

但这是不正确的
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
" g' J* X6 N7 Z% |% C6 u并不是   (a+bx)*C(x)   的导数除以   x   的导数。

6 ]/ `8 `, U3 \" }9 S0 |) V+ P5 `. [no "x" in that position.

gnome-kde

d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
....
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

+ R# c, R6 q; E这样算混淆了微分和导数的概念。

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
/ O! s! [5 Q! u, U- n$ [: 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)

- E6 a, C: M4 |* X- d/ d" [[ Last edited by gnome-kde on 2005-4-6 at 08:31 PM ]

十年移民路_

Originally posted by gnome-kde at 2005-4-6 20:30:
这样算混淆了微分和导数的概念。
/ {7 A4 w6 [+ {# ~+ f/ o- H
, J, Y/ ]$ F* T: i7 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
; D. B1 p; h" |9 p* q' R4 x
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

这样算混淆了微分和导数的概念。
! r4 d9 N% M" Y
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

" N0 d8 e0 ~3 U0 O* oAh, you made a mistake: it should be:
: r8 s5 o6 @# u+ h: \
d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
  @( F. H" N3 v. y+ T: D, X

$ ]# a% `3 [" L9 i: @5 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)

/ B& n7 E) Z) ^/ j* [for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

h(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
这样算混淆了微分和导数的概念。
( ?+ E* }, p( T2 J' V  M& N
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
) X7 z; A. T; l
Ah, you made a mistake: it should be:
9 r- L& v+ C# c2 x$ u+ E$ T" R
d(f(x)g(x)) ...

6 y2 X; u" _* s# d5 B; p. v
Yes, there should not be " ' " after "f" and "g", it is a typo. :D
5 ]8 a( ^) p8 }  A
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

十年移民路_

Originally posted by gnome-kde at 2005-4-6 20:51:
% J% {4 W& _* l- I' S0 q7 YYes, there should not be " ' " after "f" and "g", it is a typo. :D
: o2 W1 @& g2 ~1 d( m- |( n: N
9 a8 o# E2 b' `1 ?But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

2 q5 j0 r; E7 P9 I% b- B
) p! v' r& z, H1 t( ~6 xThanks, I am happy here to discuss math.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-5 08:41 PM:
) e4 |" T" g$ w# t  h# P: H
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 V& M0 J% y. ^% N1 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 ...

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？

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-6 23:24:
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。