数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！

8 e1 k! }0 V9 q4 J9 \, [9 n5 Gd [(a+bx) ] /dx = -kc +s
$ f2 v8 a+ Q+ Z8 zwhere: 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！
% W: w# H* c/ |, G+ E. E8 S
d [(a+bx) ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = ?
! _2 u4 L; J) d+ @( ~% i
, ]0 J1 T8 }. {# U多谢了！

3 f/ z0 l# m  C, K1 c/ z' Hit is too easy:

7 ~9 k3 S& Q8 n4 A, |/ Nd(a+bx)/dx = b
8 Z3 ]$ P0 U' o5 f$ y9 P9 f
so

3 P4 r# v0 [! d& q2 w) Tb=-kc+s
$ h1 F  M) b. X$ J
- b% [$ [0 l) b. Q  Nfinally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
& ]) L8 E% Q8 X3 H  Xit is too easy:

7 G/ I& |3 e# e- ~1 w' m" i0 _d(a+bx)/dx = b

" _# |; v* j8 m2 e3 E" n$ [8 a5 Gso
- N2 z- [4 O/ s+ |% w
2 y4 T& ]7 y8 [/ Sb=-kc+s
: C+ n( O8 ^0 B/ o
5 p1 D0 H& `% t% q5 Yfinally:  c= (s-b)/k

; Q) N9 l) j: d6 r/ t+ c$ i4 f- Q
4 F# Z; A# {. L' g: \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！

) N7 z+ W- L7 f  d. u% m0 Id [(a+bx) ] /dx = -kc +s
- v) A5 q% J1 `6 i8 `/ N. B' Wwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
7 A- Q: b. X( O$ A3 f5 ]
* f: P4 F) {  [0 M7 g多谢了！

5 m% R9 L! U) S" }[ Last edited by 醉酒 ...

& h: r/ c( V7 X4 |( U0 @3 xsorry, did you post another question? your statement is still not clear.
. m8 k1 O& @$ g* b6 T% J3 `3 q# E
a, b, s, k are  constants? c means c(x) ?
0 |: Q2 ?9 a- r+ Z% g4 g8 T
8 ?% ]( @( m) Z# D# q6 x! Qplease 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.

( \# y" b- K9 J6 l: A0 J# qa, b, s, k are  constants? c means c(x) ?
0 R" P% a' G- o) M& W2 \0 a$ \5 h
please tell me.

对不起，写错了，忘了变量c,应该是
" T, s: }* r& k8 E5 {d [(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:
- `, U# ?; m3 ~+ i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:

/ T1 i: g; [. a; z& Jd { (a+bx) C(x)}/dx = -k C(x) + s

where a, b, s are constants, you want to know what is C(x) ?
$ t3 }( z/ x8 _1 `6 v! G
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:
7 I6 s3 Y" B! S# `
d { (a+bx) C(x)}/dx = -k C(x) + s
+ z! q2 ]( |2 J- E5 f. y
where a, b, s are constants, you want to know what is C(x) ?
$ e/ b# b1 D9 p' f
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.
$ B$ o2 v/ n% ?5 f7 A! n
4 M! i& Y$ e  Y) R: K. e% S: r
procedure:

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

bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.

7 Z. A; N% N3 Q3 `; Z( H3 m(a+bx) dC(x)/dx  = -(k+b)C(x) +s
2 a1 d. ?3 j) ?! ~' G! ~: y- R
& z6 F' ]6 ^$ O  H- B) _* F
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
. w$ S* N& e% L3 g, `  f$ htherefore:

+ i& D; f7 x0 w5 W7 N, d' T{(a+bx)/K} dY(x)/dx=Y(x)
% W) X0 i3 }& c( s6 U8 [3 c
1 y! K/ K: c4 Q% G3 H$ j5 v0 `8 sfrom here, we can get:

. x! Z1 d/ Q9 G# X7 J$ WdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)

& \! j3 J# L" V, j+ S1 N& K! z0 _( [) bso that:   ln Y(x) =( K/b) ln(a+bx)

: [" f1 N: Y4 t+ a& N, ythis means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
% Z& ?/ W% M. O4 E7 y5 u
: o. v4 v% R0 o) K3 E-(k+b)C(x)+s = (a+bx)^(k/b+1)

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！

( R3 c/ u- a6 c- {. K0 T9 G/ Y
; F0 A, p( x1 w, z0 _6 I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。
# L  [  x+ J6 B; i- Y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。
. ^9 g' j9 R: m) l您说：

d{(a+bx)*C(x)}/dx =-k C(x) + s
  \- I( q. {- I! }* \$ R0 Lso:
* X( Q6 C9 {  g# c/ A6 r4 m" `& WbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

2 }! w( c' ?: P* @/ E. O: U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
+ {  \9 z$ R9 [. T& z4 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
9 y5 z# q0 q( d8 V, t4 kx' 意思是x的导数，y'意思是y的导数。分母是y的平方。
: p. l0 A+ J% i% d+ b因此：
& p: |( Z# ^' G  w* q# E
$ U6 l* i: b4 G( _& K' k+ R3 ld{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
+ ]+ w% B) {3 b2 j4 r! A1 G最右边是x的平方。

所以您解的有问题。
$ {( V3 b6 I, x6 A) |0 I5 U1 }0 s1 W1 v
0 H1 ?- |$ _! ^# o/ @8 P9 r+ n[ 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:
5 v* j, y5 l  ]9楼的答案有点问题吧。
5 q6 l7 }  o* ?5 `9 m% k. ?您说：
  b0 s+ |4 V6 N( X1 Q6 i4 v
9 N! J! r' [$ p3 S: ~d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

7 V" ]# n9 _! ]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
d{(a+bx)* ...

+ ^, z: |! H- W1 |
Please note here:

6 \8 w* p, \2 I& z% `9 rd{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
' |  q2 s1 W0 t- y1 Zit 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
. K3 ]" G/ ^4 y2 E5 Y$ P6 r$ {并不是   (a+bx)*C(x)   的导数除以   x   的导数。
/ Y' O8 R# g/ D: N8 I( f
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
....
1 V" o' {" w, u6 O% t1 [+ m/ p7 x+ ~d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

8 \  G& q& a" ]; c$ 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

% E' u* p9 G( }( aIf 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:
9 U6 j7 @. f5 D& c+ w这样算混淆了微分和导数的概念。
- J4 B% e; Y: f' \$ n, W3 f6 b7 c7 y( D, ^
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 ...

' i8 \6 M' b4 h3 j$ [! X这样算混淆了微分和导数的概念。
& s! P/ B! n! a2 ^  Q
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) T5 ], J, M2 U
Ah, you made a mistake: it should be:

d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx

2 |% L/ u: q! a. y: [
1 p- i+ Z& y; f2 BIf 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:
" }2 C: m; J6 \9 V
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。
) w! z  d* D+ T5 s* @2 l9 ^% `
) x/ F& I5 y! a9 u& \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 [$ A; h: G7 Q
8 J& g/ h2 j* r/ d8 t: ]- g  U3 XAh, you made a mistake: it should be:
4 ]2 U& Z' p- S; B2 E; ~
d(f(x)g(x)) ...

, e& @, |) b( [, p; s& L
Yes, there should not be " ' " after "f" and "g", it is a typo. :D

3 S9 S2 M9 Y" A9 |$ VBut 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:
0 s! E4 u: @9 vYes, there should not be " ' " after "f" and "g", it is a typo. :D

# w. p- I# H  a% ]But for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

! r! s8 H2 i2 f# w( ^+ WThanks, 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:
$ ^/ S& {, D) x2 N! o
+ i6 f' P/ R  y) I  ]  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。
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。