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?

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！
3 Q8 q3 P/ {4 ~9 x' \1 c
( a- Z+ h. p% P3 U) Jd [(a+bx) ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = ?
: m) W+ t( H3 P* E" m
* m0 r6 M" B' _; P多谢了！

it is too easy:

d(a+bx)/dx = b
. e4 L/ ^1 T* ?9 x2 {$ y, l/ q
+ C, Q4 Z0 d! k1 p. p) G8 z$ lso

b=-kc+s
$ p. ^- Q& G. Q% x9 E, r
finally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
it is too easy:

) c. ^/ I9 f3 @d(a+bx)/dx = b
9 e# ]7 |% z) o' N
& G) N4 r: x: `. u$ g+ j5 _so

b=-kc+s

" t$ _, }7 _* H+ S' x: ]finally:  c= (s-b)/k

/ B  y, L! O- P. v4 }- \) p
5 B. u6 c- v* xthe 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:
$ ], `  ^7 v) Q, s请教大家一道微分题，多谢！
5 G4 ]- M- }- y+ s9 Y9 g
. U/ T3 T1 A, J$ Xd [(a+bx) ] /dx = -kc +s
! _  y4 S1 X6 a8 [8 d6 awhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
3 F' b! t/ C" w
% G3 d. ^( L! S* F" l多谢了！

[ Last edited by 醉酒 ...

! W; y" R* }! usorry, did you post another question? your statement is still not clear.
/ N( R) @7 i; l  y
a, b, s, k are  constants? c means c(x) ?

please tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
+ N; s+ F7 R9 Nsorry, did you post another question? your statement is still not clear.

# R+ `  _+ b9 g7 La, b, s, k are  constants? c means c(x) ?

5 M$ o( n) [" f6 k4 {+ Kplease tell me.

对不起，写错了，忘了变量c,应该是
, H$ @) h0 b6 H* \2 Od [(a+bx) c] /dx = -kc +s
+ k4 L- G* Q8 y7 N- k
多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
9 T; f5 M; D- A8 P对不起，写错了，忘了变量c,应该是
1 z/ ]* C. c0 ~9 ud [(a+bx) c] /dx = -kc +s
1 Q" N6 K. U/ L2 U' W+ @+ Y+ Z
; M" K8 ~9 W' X. R. O  u( b多谢

  e1 u) e" F$ ]) }: {+ Q" e' Kdo you mean it is:

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

- A5 O! d8 ^1 M1 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 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:
6 j3 T* f. W7 K% N7 \6 u# E8 l: ]
d { (a+bx) C(x)}/dx = -k C(x) + s
+ E& g( F; m1 j# [3 O* _
8 V3 U6 m9 ?) X8 e1 Vwhere a, b, s are constants, you want to know what is C(x) ?

4 x4 K% F4 Y$ i, p& @3 Lif so, this needs to solve the differential equation. please comfirm it fi ...

" r, m" R" @  i4 m( q
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:

) k- Z$ z' p' w# e( _+ CC(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.

* t, L" @0 u$ C' Z
procedure:
" v9 O8 U1 s# t& H! X
4 u. @  b; x9 V+ aFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
' A; T# J' |$ U- c2 e/ S
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
, c5 R/ t$ }( B- w7 B, _
6 W& N, W8 f8 G/ N2 L(a+bx) dC(x)/dx  = -(k+b)C(x) +s
/ u: R: A# P% a" [

: @" ?; b& x- d: M# ~6 M. U% Lintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
: J9 \) b6 ^5 y8 ^$ s, S  Kwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
- c8 y4 ]" I5 o; h+ e% ?therefore:
. }- Y, X  L7 \
7 t2 T' Y- y/ ~8 X) N# g2 B{(a+bx)/K} dY(x)/dx=Y(x)
6 }& ~9 O! E$ Z5 z/ _
from here, we can get:
. D9 j- {' S7 J6 X  y' ~
: l. Y4 ~# C3 A! _5 f2 IdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)
' j6 K7 Q4 g4 p% e1 T4 {
; {- Y! a1 X. }) Oso that:   ln Y(x) =( K/b) ln(a+bx)

. y' \; {' Q- M! Pthis means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:

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

finally:
2 i+ q* ?3 T4 U( |$ v) X4 m
% P+ X' Z2 ?: [; n& l2 yC(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:
$ c1 B7 r3 z; h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。
( K6 P& H! x( e+ h9 [- |6 E" s您说：

. d/ P2 g/ f: _d{(a+bx)*C(x)}/dx =-k C(x) + s
. F* K- b  F5 e. v* f: R2 E! S1 `so:
0 D& p- `8 l7 @9 q- j: ubC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

2 T& a0 i  ?1 m4 _( W' A$ Y也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
' Z* k1 W* [9 \$ v" I* Y3 g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
x' 意思是x的导数，y'意思是y的导数。分母是y的平方。
  R+ V& `& }% H3 N4 y$ `1 ]因此：
$ u0 o* u, W3 n% C
6 u2 Z; I2 c9 e  Ad{(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。

7 p" v! }% k" }( V所以您解的有问题。
8 H+ b: B7 W5 W. F! D& Y0 G4 \2 Q
[ 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:
  ^& i& T6 ?8 x2 y5 e+ h9楼的答案有点问题吧。
5 C0 n; s1 K% x' K; [您说：

d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
6 g$ w. [1 ]3 z3 l6 _1 ]5 B
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
7 x: c" a3 `1 L- `! V但这是不正确的
d{(a+bx)* ...

# O" m, r2 A9 U7 |$ o4 J6 c: d. S
. ^; }  m/ I# K9 s+ r7 _, TPlease note here:
, H0 V/ W# ?: V1 R9 @7 S
9 s2 T9 k1 w3 o# Bd{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
6 P- P! c4 ^8 Y  Z% T9 k, n2 k8 \it is a multiple, not division. Right ?

十年移民路_

但这是不正确的
6 k0 H# A: @. B4 P& b7 J( nd{(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。
7 s$ n8 d3 D& z  L0 {
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

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

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)

[ 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:
; x. c, S; ?9 C: |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

- _! c/ }3 f' m0 ~1 KIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

& \" C) Z, {; [7 j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
0 y) e# L0 _3 D; S# _6 i) D$ Y+ D0 G
/ \( V1 b9 j( a* a& @: T$ vAh, you made a mistake: it should be:
4 ^4 C1 Q8 B. J" m% M2 r& d
" y: |; K& ?. {9 M" z, B. _! Wd(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx

6 ?7 k' i9 n2 R4 @
7 Z: k6 p/ A9 ?3 Q9 C0 l' K! A: F& qIf 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)

2 M& {9 q1 ~8 G7 Z  M) Cfor h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:
3 U1 m! q3 z6 `  o! p- Z" ^6 K+ X
& o: z8 {/ W6 O0 y6 a0 Dh(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
, S' j- w9 H" R$ U. Z这样算混淆了微分和导数的概念。
/ s& J; Z% K' N3 p* 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
$ L5 P& N* X  S4 w7 C( ^2 L: A
Ah, you made a mistake: it should be:

0 _+ j. a/ ^) q* H  v6 kd(f(x)g(x)) ...

( V+ O& P- G# S9 P! b0 C8 TYes, there should not be " ' " after "f" and "g", it is a typo. :D
3 W6 |( q2 L+ _: M0 h
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
1 n% D1 }- z% [5 @# @  I8 n3 b
2 p* |! _( f/ m* IBut for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

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:

& R: G# l4 r: E: W/ w5 M9 e" j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。
7 e+ K$ |7 m: F3 I" A: A9 r: z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 ...

- Y$ f' K" R- d( O. l) [2 w
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:
/ t- J+ t+ h+ I0 ?3 t* f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。