数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！

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?
* M4 I' @' w/ y9 p4 f8 k$ F$ O6 e
多谢了！
4 h6 L/ i% J  A
[ 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！

d [(a+bx) ] /dx = -kc +s
' `& u! l3 \3 S( Q, L- E. p7 kwhere: only x and c are unknown, others are all known,  requiire c = ?
, P1 Y! X, ?" W: K( x+ ^
* M& b1 @( V! L) e) e6 j, P, ?多谢了！

it is too easy:

$ k8 _+ p! H* ]) gd(a+bx)/dx = b
1 `' w- Y0 y: Q- \; d" u% X
5 \, N1 x( w/ m4 I( f! h! V9 ?( @so
* d7 w4 D+ ?" W. b5 z8 {
b=-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:
  c/ \8 {& [5 I5 Sit is too easy:

d(a+bx)/dx = b

) N  E: b1 @- S" ~so
# {0 }* q7 Y4 }" T- I5 f( B
. U: N. b# t* R/ V( y. e8 `b=-kc+s
* K9 ]7 r" m7 C( [9 d0 m
finally:  c= (s-b)/k

, O1 B$ |! b) _" w# p; E
4 f7 z) n7 `& ~. W% F( C& ]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！
  L+ q7 @; J  L2 j2 [- p" p4 [/ I
d [(a+bx) ] /dx = -kc +s
( S  G! u9 T4 N: ?! j3 C# xwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

5 E, t; G2 k5 x  w4 }4 |1 o多谢了！

2 L( U$ S+ D& l/ ~[ Last edited by 醉酒 ...

sorry, did you post another question? your statement is still not clear.
% f, i! O& Q& _2 w: M; K
( Z  q* a0 J. {9 L+ N5 ya, b, s, k are  constants? c means c(x) ?
. H9 S- ?' P3 i$ f5 o
please tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
, {. y0 d/ Y3 a  `% R% N" Y3 Asorry, did you post another question? your statement is still not clear.

a, b, s, k are  constants? c means c(x) ?

please tell me.

对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s
; e5 f8 N3 N0 L. Y! u7 C
# J( A7 ^, |! A3 Z2 M0 M$ W多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s

多谢

do you mean it is:
' e% a$ a# b  |+ k1 P2 d! D9 H
d { (a+bx) C(x)}/dx = -k C(x) + s

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

( v) E3 r) ^) W1 Aif 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:

% O2 O& j8 y6 c! Hd { (a+bx) C(x)}/dx = -k C(x) + s
- l, [' f# W! W& q
where a, b, s are constants, you want to know what is C(x) ?

7 K. A+ P, c  @: j- _) X5 z6 L! b/ Uif 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.
( j( W. K5 }, N1 h' }6 c
7 L- b2 V* Y- M% B
% R" e1 p/ w/ g+ ]procedure:
0 P% ~' n' Y+ F# R. X  f
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
7 C1 K" n6 d" Z* x  ~9 v
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
. I9 K/ V, e2 j# Ti.e.
: p8 i1 K' ?3 l3 h; ?
" \$ i* s! r8 o7 v* X4 S# H, [(a+bx) dC(x)/dx  = -(k+b)C(x) +s

, ]4 X3 c; a3 b- `9 X
introduce 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
0 i* d5 ^% D6 {- ^: E8 U( ]  Vtherefore:

{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:

dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)

) o5 x! e3 c8 f7 W# d* A, eso that:   ln Y(x) =( K/b) ln(a+bx)

  |: o' ^. H3 U! }4 h& b0 Qthis means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
- @+ b: `5 a6 L0 X
) y% a- U- x+ K1 g4 p-(k+b)C(x)+s = (a+bx)^(k/b+1)

* a" L8 H* ]5 F+ t- P- P: xfinally:
6 E3 G. M/ I& |* n
6 X* L" m1 }* v% z$ L4 jC(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:
$ O/ Y4 p/ r! E( S1 S9 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！

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。
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。
; \6 w2 ]$ C- }0 i( {您说：
1 Q* B2 L, {; P* y) }; \4 J  s
d{(a+bx)*C(x)}/dx =-k C(x) + s
& |8 d6 k" E7 N7 q+ l$ i% ^* `! ]so:
$ Q# G+ r& S) JbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

4 l) r  c- u* S  o+ n# K+ p& Z也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
6 O; n$ l3 O9 D: |7 w# L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
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
, O7 c6 Z; ~/ p" p6 ~- Ox' 意思是x的导数，y'意思是y的导数。分母是y的平方。
因此：
7 k3 ~+ F1 f+ S8 h% j  j
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。

所以您解的有问题。

9 I. g- i( y5 h  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。
& P( y* Y7 i  |9 A* u1 i您说：

0 }$ o" t4 \0 g. k5 o$ [d{(a+bx)*C(x)}/dx =-k C(x) + s
+ i% Z" q7 ^: d' M, U) H) H/ tso:
* b4 C1 K4 o1 K& @6 |% ]bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
; `& T3 |4 K8 D2 M7 M0 ]
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
1 }+ \, x# @3 \! r8 H: h但这是不正确的
! _. D3 Y- D! Y8 _/ U. B+ p. b. Gd{(a+bx)* ...

. @3 x3 ~5 r9 h
Please note here:
! B) r7 Y$ D+ a
d{(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
. N& I  o4 y# w$ n6 Q2 y! G并不是   (a+bx)*C(x)   的导数除以   x   的导数。
) _8 \$ G8 ?- o/ j# Z+ _
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
....
4 o0 \7 V9 b5 C) @% Z. V$ wd{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

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

* f, [- V/ e9 Q0 z1 g/ S, `9 PIf 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)
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! U6 @% m; o( G4 R[ 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:
! |7 _! }: U2 Y, Z' z, t这样算混淆了微分和导数的概念。
" o/ v1 w' t  a! G( Q% F- ^. Z
& W3 d. y# P% D) a! 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

If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

  C% @/ z1 d. i- c& I4 D' P
$ ^& T- K9 W/ E) l  w. w这样算混淆了微分和导数的概念。
, U' I- [5 C8 B; H. v# E8 v4 w
$ e* V( f. \# G$ m! g  z- C1 D: I+ 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
, V" a) }; ], u8 s1 }$ E+ b) a" l
) {7 U- k" X6 z* WAh, you made a mistake: it should be:
/ |- R$ L& o% Q. J& Y
d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
2 G& M; I* ~6 R8 m6 ^2 M

4 h' X( N3 M& ]# Y& W# c  \9 A7 MIf 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)

( X0 h9 k2 e& S, \0 h1 z+ f7 N+ M9 ]for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

1 Q+ n9 w) h; Y. D' @5 ^h(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
0 y& m( n+ O/ K. ~& P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

2 k) t, d- S4 R( \$ P: U& ^Ah, you made a mistake: it should be:

3 l( }: P6 K# a# wd(f(x)g(x)) ...

Yes, there should not be " ' " after "f" and "g", it is a typo. :D
. p# z  r% |) s5 A: f  z" g( {
1 V$ m# C7 @0 c- wBut 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:
Yes, there should not be " ' " after "f" and "g", it is a typo. :D
& }( }2 A2 Q% ?* ~9 r) f
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

$ k+ h7 y+ X$ v3 P' B
/ V5 L3 [  ?$ a4 [7 ]Thanks, 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:
" v3 n) c; L0 w6 P: Z/ m4 }# H' 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。
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 ...

( D. i8 \/ ]9 I) F- [( k9 Z. n; L; R
7 r' d1 a1 Z4 ?  O' ^: _# 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？

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？

$ U$ r' R( H- X$ ~0 P
* U; A1 t, o/ K5 N- U- k5 Y8 W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。