数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！
+ t; N! S, v; F, V' J
6 y7 u3 e) B2 n! U: hd [(a+bx) ] /dx = -kc +s
2 l+ R; _/ R2 v# X+ O! kwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！
' A" e. ^+ \8 w7 n, I
[ 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:
/ I6 L7 t% X8 I请教大家一道微分题，多谢！
  p. u( v* Y7 I* R- I
d [(a+bx) ] /dx = -kc +s
: c! l) @8 J9 l" ~' X6 q& awhere: only x and c are unknown, others are all known,  requiire c = ?

- Q: A6 d) m) O+ e. P6 [& A, ^多谢了！

it is too easy:
9 f! S) K' a+ ^, U7 z' W: f
d(a+bx)/dx = b

2 [" f3 g# Z) t7 {so

2 I  O% y7 N- i" H) B. V1 nb=-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:
- H6 a, u- T2 j6 c' Kit is too easy:
" e# t+ B0 o! A1 B& c
d(a+bx)/dx = b

# V1 Q6 Y4 q9 v) _4 H* O7 fso
; M9 b$ C; l6 k8 Q
b=-kc+s
0 O. O# M0 M$ Z1 J
finally:  c= (s-b)/k

6 n% `9 }- F& S- fthe 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！
5 G& m5 j/ w) X5 |( ~
1 M- \- O* h2 V7 X: id [(a+bx) ] /dx = -kc +s
% y" z; o, l6 O. E5 b& fwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
8 B: S0 r% Q: L6 r
3 t6 ^, e- m9 P& |% Z' `多谢了！

[ Last edited by 醉酒 ...

sorry, did you post another question? your statement is still not clear.

4 Q  M8 I$ H! K% @2 _. Ka, b, s, k are  constants? c means c(x) ?

% F4 q/ h& E9 Z+ F9 ~please tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
; O2 U) F; G* W8 k" h0 `5 {! M# ?3 O! Gsorry, did you post another question? your statement is still not clear.
4 c4 q' ^, N2 S* v0 l
% K0 y/ Q- z6 X/ ~/ x; Xa, b, s, k are  constants? c means c(x) ?
2 q* V: b* l" m8 v
please tell me.

% C( J( o9 f8 T2 |' ^" v, f' e对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s

; v" i* R- y( }7 {8 }( O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

& G/ h# B% i5 G  R多谢

0 J# C3 g2 T4 s5 A9 P
3 d9 C3 l0 {% t3 kdo you mean it is:
3 M- P" U  {" L2 u+ h# V! }
d { (a+bx) C(x)}/dx = -k C(x) + s

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:
$ L5 @' v, d7 a7 z0 tdo you mean it is:

d { (a+bx) C(x)}/dx = -k C(x) + s
' z# g+ r: g) l% I/ [
& _3 K: H5 P' |6 q4 J% p+ ?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 fi ...

1 x* b2 X+ j# A# g# s" h! r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.


  ]- T. ~  I/ {& z. F: mprocedure:
$ ]3 g: D# R' X: o$ ?' S) r$ u7 g
# [0 ?% M  F- f9 U1 L5 kFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
0 T( `! R+ y( o0 Kso:

- n% `! j* M) L$ T( q( `bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
6 O# A+ I4 D+ j7 s. K8 [0 v. d
6 {7 e  t! c0 _. d" K. v/ a$ N6 e(a+bx) dC(x)/dx  = -(k+b)C(x) +s


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:
; ?8 P8 h- G1 K: g6 R& E5 G$ k
( E  w) D6 W- b6 ]' ~$ e6 C. {{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:

4 B- G0 B% c  hdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)
9 I9 }! w; Q1 N! {# i% u
so that:   ln Y(x) =( K/b) ln(a+bx)
6 y) _. n, b0 c( ^3 ]( Y: G* q& J
this means:  Y(x) = (a+bx)^(K/b)
& w8 Z/ s8 K: y$ T) _7 nby using early transform, we can have:

$ w: U5 y  l( P-(k+b)C(x)+s = (a+bx)^(k/b+1)
" v1 E2 j$ J0 d" `7 b! }* G# n, \
finally:
. w& R, E% z7 a6 l, x
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:
+ ]6 e& x9 F1 N- N1 G( J5 _# R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！

3 y* d+ R# c, F% h+ c$ I" o7 s9 ^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。
8 H4 P) c% E; s& S+ o3 x1 x  y您说：
- i2 X+ s0 x; K( T8 ?  h3 o  {
8 ~! [, X5 G" ^$ ed{(a+bx)*C(x)}/dx =-k C(x) + s
' }* M  k- z4 P8 ?; J: D" Vso:
) o# G# l& J$ e0 kbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
: B4 I0 T" S. v7 {
1 h6 R4 `( f4 I- Z也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
2 i) R3 o* ?& m但这是不正确的
* P- Z$ \( g& ^! |) ^& p, p3 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。
8 _1 e5 h9 g! Z" \- n5 m$ I& l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。
( o- u" B& f- K# r; ]' {因此：
' z7 a4 O. N3 |9 a
( y7 ^" W5 m+ f2 b( @/ Td{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
* c/ j$ a4 Y' v/ _' m* D# ]最右边是x的平方。
4 z" q6 l3 f. U$ M1 s
$ Z- M9 w* U( g; |: V2 m所以您解的有问题。
1 I+ G8 G: s# J! 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:
9楼的答案有点问题吧。
您说：

! s2 S' I6 O* h2 T4 E/ A, d( ud{(a+bx)*C(x)}/dx =-k C(x) + s
3 p" A" B4 q: ]2 D0 P, fso:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
, n, t/ N& M4 U( J+ }
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)* ...

- |$ g  H5 ^* i
Please note here:
% G. l6 i) [  g7 w  D6 n: ^
% @9 E. p6 S. r6 m+ b0 T* u7 Md{(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 h* p& P9 l' vd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
% E! ?; i/ S! h并不是   (a+bx)*C(x)   的导数除以   x   的导数。
1 \5 d& m4 r" h. Y6 Q8 U7 G) l
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 |: q4 I4 p7 w8 e, F* jd{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

' p: X" P8 U) ]这样算混淆了微分和导数的概念。
2 U4 ^0 M& x4 _2 c2 F+ X
6 C* M$ z0 `& j% N0 Yd(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)
5 q. c* |- s/ \  c2 Q( M& ]
5 K' Q( |1 P/ Z' w0 m. B[ 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。
; U/ K% M! Z  f" L" [
1 @9 G# Z' b5 V2 p$ |! 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
) p) q+ L; P( H* h8 A  U
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

* P& D* y0 y5 k3 ^
4 H0 A3 e1 n9 }) e: p! _7 B, E这样算混淆了微分和导数的概念。
1 O, w4 H  d5 v0 t+ N3 j- M
$ p3 @1 [& p( @% x! P3 l% u1 n8 Ed(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 p  ]3 Y1 B. D. j6 A' ~2 T( R
9 J; m1 ]! G/ C" M* zAh, 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
3 }$ e- Y3 j4 q; W/ R; G
) F% v1 |7 V; j
$ ]! W) r! g& g  t+ d$ WIf 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)
! F. f8 ^. P% t$ ^; Z. x
' y/ r3 c( t+ efor h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

0 ]6 ]* |" H) L8 \9 Ch(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
% f) O! H$ j) A! S7 \这样算混淆了微分和导数的概念。

% s; M& y+ G; |4 Bd(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

5 D4 w: j) Y- v* cAh, you made a mistake: it should be:

d(f(x)g(x)) ...

$ v! i, a* \+ o! R5 rYes, there should not be " ' " after "f" and "g", it is a typo. :D

* X% u3 _& E4 O* `4 q1 CBut 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

But 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:

5 w. x- H' b- J1 ^0 g. l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 ...

1 Q" M5 v4 b- H. Q( d: U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:
+ r& x- C% E  _4 y* ~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？

1 X, ]) {) H1 D& E
3 v0 I! B3 n. g2 r! i. l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。