数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！

3 f) h) L7 C* M7 T% [& T! d0 ud [(a+bx) ] /dx = -kc +s
5 g- K  m! L& W" d' _where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
0 Y( N: \: L& {5 V. F
多谢了！

[ 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
where: only x and c are unknown, others are all known,  requiire c = ?

* ^' v) V% t  S# M多谢了！

( |% j- j+ y3 D. s; Y! Fit is too easy:

d(a+bx)/dx = b

so

b=-kc+s

. L, A1 q; W9 o$ s: L. zfinally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
7 r6 `% l. ~0 |$ tit is too easy:

d(a+bx)/dx = b

6 j# F" W3 Y! M% Y5 U  m- yso

b=-kc+s

finally:  c= (s-b)/k

( C$ i4 G: g: g/ {! {6 y
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！
1 F  ^- l3 j! J/ u& y# o, F$ g. K: g
+ R, i  S# R2 T5 c0 b4 pd [(a+bx) ] /dx = -kc +s
- S' u0 |# q6 H: W: E7 hwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！

[ Last edited by 醉酒 ...

3 t5 V9 K( ]* u; Bsorry, did you post another question? your statement is still not clear.
6 I3 d' L9 ?  p3 v
2 B- d. F# W; O  F& _- Qa, b, s, k are  constants? c means c(x) ?
' n4 U' ]( y# s9 l8 u3 T' X
please tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
6 P4 H+ f* D+ _/ U4 `  h/ h$ \; Tsorry, did you post another question? your statement is still not clear.

a, b, s, k are  constants? c means c(x) ?
5 F( P1 Z9 a4 J
$ h0 `+ ^4 p" z$ A% G6 v3 l! zplease tell me.

对不起，写错了，忘了变量c,应该是
  {. k$ i+ |* f2 u+ q$ ~d [(a+bx) c] /dx = -kc +s

5 {: Q- O5 L" Q3 r多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
( i; j* R" r1 ~  f8 s4 h" {对不起，写错了，忘了变量c,应该是
/ k" R: ]) j' Nd [(a+bx) c] /dx = -kc +s
1 ?1 ?3 {" @  ?6 j2 R9 p
" A  o% S' I  ~# [2 e多谢

do you mean it is:
: S$ E4 ]# n4 I4 q9 E1 {1 ]
d { (a+bx) C(x)}/dx = -k C(x) + s

( z4 S3 a% ]: E: A. @where 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 first.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-4 06:36 PM:
! l4 P: U$ o$ z' edo you mean it is:
% @8 k: s6 x: V# ]/ l
. e' o! @( t! a  v$ W. ?d { (a+bx) C(x)}/dx = -k C(x) + s

2 Q3 N0 r! u  G) U8 wwhere a, b, s are constants, you want to know what is C(x) ?
6 b1 c) l) j6 W1 r$ P; K" d
8 b9 E' f/ l/ Zif so, this needs to solve the differential equation. please comfirm it fi ...

; R; ^  H2 a  x7 ]8 H) t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.

8 ^' h: z: r0 n: g/ l
procedure:

% q; E1 m. E, C4 uFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
- z) b) F# e8 Y+ S4 zso:

bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
$ A9 Y1 T9 G; s- U3 E3 i( \- ]i.e.
" O( h9 M4 ]9 u  m
(a+bx) dC(x)/dx  = -(k+b)C(x) +s

, H) \# [& X- R
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
therefore:
9 z9 R: g2 a0 W: _5 L$ A3 K5 X0 c
& V( Q4 P: K5 g' @$ P8 o2 r{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:

: d% [0 a- l8 |* T, h$ `" HdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)

8 y  z7 p7 m& v4 `8 Cso that:   ln Y(x) =( K/b) ln(a+bx)
+ g5 k$ v9 e, A
% e3 A" _! k1 |: s: y$ T. Sthis means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:

! c, ~; W) c" |9 ?( K-(k+b)C(x)+s = (a+bx)^(k/b+1)
" }' N# z# M6 }2 ?* T/ `& s4 X
, M# i, L' y. X- I' B* ofinally:

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:
. V4 Q# U! C# A+ H) u* o& W9 S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！

" |( M2 E! _/ [" d% z5 @8 Z( 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。
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：

' W1 P# }# S8 ^: E8 Ud{(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
, ~, t- ]" V: b. x" j( z) ^/ [但这是不正确的
( g0 c2 o. K+ h4 z3 Sd{(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。
! J8 i: t% S: E2 |  B- F比如说   dx/dy = (x'y-xy')/y2
/ A3 S5 \5 q- n( H% zx' 意思是x的导数，y'意思是y的导数。分母是y的平方。
1 j5 C3 |8 I" N因此：
& U+ d$ b$ Y3 T" N
2 z# \, N, i! ud{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
$ D6 W8 J% q. E; G* \7 u, C0 C最右边是x的平方。

* G" B" ~1 U0 |" m2 P5 D所以您解的有问题。
" F! M# K, F: i# g
[ 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:
1 i! j6 I$ C; `( z9楼的答案有点问题吧。
您说：
% b0 ^$ u, ]4 T0 k& G; x
8 f& D7 q/ n4 Y* a9 Qd{(a+bx)*C(x)}/dx =-k C(x) + s
8 i. p6 O( k' p9 n: Y% k& z3 Vso:
  q, E" n( f/ l- O. w  rbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

9 _7 k0 y8 q, _! r1 m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
* _; T- G8 S  H* p8 L$ |+ Qd{(a+bx)* ...

Please note here:

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
8 o1 K0 D, e0 _/ `- R0 ld{(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。
; i; X# q) j+ W5 _! i
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
7 c8 t" i# Q, d- Z6 A; p# a....
d{(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。

& L% ~$ |+ F  nd(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
4 p# d* s; T7 O* P3 }  D% I
. K9 u" k0 B" d- s6 c. EIf 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)
; \# s+ M7 j( O! r" w! [
. {8 \/ b8 ]+ B8 z9 f! u& x: {1 h[ 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。
- P6 x- ^4 k, a6 \8 `
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
/ m8 r; N( P' p) p+ W
7 }3 B# {8 _" Y- wIf 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。

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 E6 w2 x: s1 P. P: s, C
# P( x' j3 d0 SAh, you made a mistake: it should be:

: K6 d( T! {( G# [1 @) o; l& nd(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx


% C6 E# M. }# A9 J- oIf 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)

for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:
/ R! V9 o. ^0 q  P$ ?9 K
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。
7 W" B& }& T2 ~" k( g- v, ?2 I
. H; D+ }3 k% V! r9 M2 |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:
. a7 W3 a8 d: T
: r2 D! q& B( A1 o/ c/ Cd(f(x)g(x)) ...

1 V# n9 x; x. d8 ^% M8 Q* h
7 {% v" H8 m5 X: S; ^: q4 y, Y2 VYes, there should not be " ' " after "f" and "g", it is a typo. :D

# T. ~! L$ z- r, X; B: [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
( q% D6 e& W8 A7 t/ ?
But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

7 w: l( B. O5 a3 Y9 {- w; u% W
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:

& \9 @6 @  W, x! n, ?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。
; ]( e1 |! N1 L$ o& T7 R- v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？

" ~7 Y. x' @3 V5 z
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。