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
$ _3 m. c9 j& s& X& b2 g& M0 w- Dwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
4 X* b2 X$ E  B
多谢了！
4 C+ j  k2 [; D# ^1 Q4 p* [
7 A: G) c- W$ K% |. M* a+ r[ 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:
" E! b) H- V% W2 j) a1 y+ z请教大家一道微分题，多谢！
7 f1 S( W- L* G- s! E
, S  w; ^# w! K+ Wd [(a+bx) ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = ?
0 F* n% f4 ]) Q) B
多谢了！

" Z- J2 e1 X7 k, @0 P
4 H/ E6 B  K6 X( N; Rit is too easy:
+ `+ L% U: V# ?7 s/ `# n
d(a+bx)/dx = b

' q* I' p8 M- ]+ Pso

- S# \7 S4 a3 Y" F8 A7 B' ub=-kc+s
+ q# Q% x. w+ [9 c
4 [& H' X/ I; _: y' D, Ifinally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
8 {/ x) ?( a* q" v! y" iit is too easy:

d(a+bx)/dx = b

( K, N  e  N; |0 e8 ]3 d+ b4 }3 tso
) J" h9 U4 C7 y- J
b=-kc+s

. I  P# G. |6 l, |* W  ], V7 z7 pfinally:  c= (s-b)/k

" s0 r$ f# g" u# N3 n( P
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:
/ M2 J+ N6 U0 V% I* h+ ?( [/ J4 X请教大家一道微分题，多谢！

d [(a+bx) ] /dx = -kc +s
+ C. a) u0 b; H% Vwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

6 z! p& q9 \7 @多谢了！
( [4 m4 j8 h! s9 O
; j9 i/ e: @) q[ Last edited by 醉酒 ...

/ [8 D+ k# T4 j9 @" ]+ b; w9 L7 |
2 N0 ~0 x- U  C6 v; ksorry, did you post another question? your statement is still not clear.

# q3 }* b$ ]9 d, |7 X" {a, b, s, k are  constants? c means c(x) ?

7 `4 y  W  S: Dplease tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
8 v+ w. }$ Z0 t1 G; Q7 ^! w; esorry, did you post another question? your statement is still not clear.
% {/ _5 p1 X9 m" [; C) K
a, b, s, k are  constants? c means c(x) ?

please tell me.

6 E9 w9 R3 g! S对不起，写错了，忘了变量c,应该是
% q) j% `7 G: Y( M4 ?% nd [(a+bx) c] /dx = -kc +s

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
' x, I( U) N/ i7 A9 Y2 @, e0 I& wd [(a+bx) c] /dx = -kc +s
, q2 {& e6 n$ _" F
多谢

do 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) ?

0 K; Y) j- [; I3 ]% b# f9 gif 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:
( O. y( [& f) I- B( ndo you mean it is:

d { (a+bx) C(x)}/dx = -k C(x) + s
5 g" U5 V4 t1 x" K& B  k, u6 p% g
where a, b, s are constants, you want to know what is C(x) ?
4 D' J# T4 s: [% M- u% c- j0 Y4 p
  t1 K" I9 V1 Z9 E9 \) Fif so, this needs to solve the differential equation. please comfirm it fi ...

' o8 s" F! i: G! T7 M
* N! ]% V$ w  J  \' H  U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:
2 I5 {" v* z. o) o
9 ^3 a3 G7 ^$ S. VC(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.


procedure:

From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
+ l! z8 e) q! a( Yso:

% ]3 H* ?5 ?. V1 g: _+ dbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
; v2 e* o3 n9 E+ h$ L) c" J
7 ^' T. H. u* b& C! H(a+bx) dC(x)/dx  = -(k+b)C(x) +s


& f4 T$ l1 t& g6 Xintroduce 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:
; w. k) q# O, w9 U# M+ L0 U
{(a+bx)/K} dY(x)/dx=Y(x)

* b6 U/ t& a: hfrom here, we can get:
- t0 H4 g+ t- _. J5 [4 s
4 Z8 o) r6 L# t3 \dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)
- G7 e7 t) X: a/ {  b4 I$ X
; T4 F  u4 D( ~" U: ^8 K0 n. G. Yso that:   ln Y(x) =( K/b) ln(a+bx)
& h! y' `; c  ?+ W1 K4 C. L
this means:  Y(x) = (a+bx)^(K/b)
! ^, c9 Y8 r; W6 Lby using early transform, we can have:
2 t) l. N  h/ w4 w4 S
-(k+b)C(x)+s = (a+bx)^(k/b+1)
9 G  z9 a) m4 p  P# x
finally:

C(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:
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！

5 N$ X' A0 W) ?1 \: `4 j
$ P7 n& C5 r4 d: a7 l" B6 z- \  {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。
& v  W# W0 d5 P3 E7 q& 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）。这种题真的不是难题。

龙舞九天

9楼的答案有点问题吧。
您说：
6 y- j# a4 e, W1 t) o6 y9 T
. z3 r! l. M7 `6 Ed{(a+bx)*C(x)}/dx =-k C(x) + s
so:
' ^- q2 Q# H% l# l' X" T* BbC(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
, Z* y  [8 q6 y6 S但这是不正确的
* |( I# \* Z  M. f0 Xd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
% e2 B$ W" M1 H: w$ H# Q; F( B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
9 n8 c; G# V" H7 Y& d' b) T5 ~6 Bx' 意思是x的导数，y'意思是y的导数。分母是y的平方。
) D$ b8 `1 V- T  F因此：
$ @+ H/ i% s0 O8 d" J& ]7 _  a
- l- i& g8 @1 B/ Nd{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
: H- p' d5 ?( X0 o/ d: U4 E8 ^" w最右边是x的平方。
4 c# p% f1 K# d' @. Y: ~' T9 p
1 g6 Y! U- k( P5 l9 X所以您解的有问题。
% C2 ^6 ~0 ^$ ?% k5 q6 p
[ 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:
0 @8 }7 _! V$ O9 E9 Q7 a% o* W9楼的答案有点问题吧。
8 W6 m! J# y1 J4 a0 ~! c您说：

d{(a+bx)*C(x)}/dx =-k C(x) + s
" ]+ r% {, z$ `1 e, V9 uso:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
* J0 X) ?1 C7 n, b) t" R9 f7 K! f
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
5 z( e2 x' l) Q  Q8 L但这是不正确的
d{(a+bx)* ...

5 t* w% b/ G8 a# M8 \/ L( }1 C
+ M0 |$ ~; Q, C# A( r: O( p! L# sPlease note here:
! D6 c% j2 `+ b- d+ Q
7 b0 r& g' v( P) Jd{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
8 V2 }- b0 ?. r3 c/ cit 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
& p7 _0 c6 R8 s( ]. c8 ?并不是   (a+bx)*C(x)   的导数除以   x   的导数。

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

, H, K- L( |* w6 ?" e- g7 W6 i
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
& \# r( J/ X6 x" O( S" u
6 h  ]6 i% ~/ C, i& yIf 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( x- S3 v5 P5 n[ 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。

  E5 z' {1 N4 B7 {, Z9 F8 jd(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
+ Q0 T) t: i2 @- N- J) g
$ }- n$ l1 V" H/ q, k0 nIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

+ g. |( f8 F' H1 m( S  C  j5 m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
8 c" s! G4 K( `$ O4 m% ^- S& e
- U& d- @: r8 P1 c8 d" H3 ?Ah, you made a mistake: it should be:
( R( i3 d+ L% f
d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
: Z' H# r: t5 q7 u

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, u; }% e! D' ?# c) \
for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

! u# y5 Y. y- w2 I+ oh(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
7 Z9 i$ x7 v) a4 y4 w这样算混淆了微分和导数的概念。
4 i% t+ f$ p8 l
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

Ah, you made a mistake: it should be:

8 t7 s( P+ C% Y% e% f1 ~: pd(f(x)g(x)) ...

0 b5 P9 }, }/ s1 @2 x3 w3 hYes, there should not be " ' " after "f" and "g", it is a typo. :D
+ M4 o, [- ~# O1 f9 _2 [4 ~" n, r  Y
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:
Yes, there should not be " ' " after "f" and "g", it is a typo. :D
5 D% y' e7 b( A
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

3 g3 l5 i/ p* P/ W) b. y
: m0 t/ @/ {; a+ b. w1 K- i9 uThanks, 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:
9 w( P$ X5 S4 Z! O7 b/ q
; X" T6 b. w: D& 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。
3 B2 `: \% M% y4 H  c# v3 J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 ...

# L7 t- h9 p1 h( V$ j
3 I) f: c3 i' `/ j6 M- @' K' z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:
3 o8 T! Q+ v( l' ]# ]5 e, R! o- r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？

0 }0 Z& n- I2 V) G( O
9 C: X. b2 {/ x- b2 Z1 }% U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。