数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！
/ F% U, K; n) L7 G9 t
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?
1 K0 S- h2 J4 D! d) \
多谢了！
7 P6 f  ?- {' t4 \% ~5 f
! K0 J5 r4 `- ?3 ^8 k' c[ 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:
: X# c+ a2 J- w0 V1 w" h+ J% O4 Q* e请教大家一道微分题，多谢！
; P2 c- ?0 B# z# Q5 |2 C9 Y
! j& s7 _4 \/ ?d [(a+bx) ] /dx = -kc +s
% U3 Q8 w1 J2 c* [1 c0 |: {where: only x and c are unknown, others are all known,  requiire c = ?

多谢了！

  c4 T7 B* r6 ^$ Rit is too easy:

0 ?) H5 q$ s' i5 E& ld(a+bx)/dx = b
, C5 I/ e2 N- n- Z
so

# m  S! K1 a  d5 Jb=-kc+s
; W+ Q8 M1 a5 E( A, n# Y
8 Y( b' X) y" a6 Cfinally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
( F6 C, T, l$ a; j3 [it is too easy:
! J+ m6 }$ l+ {; h
d(a+bx)/dx = b
8 P8 a, s1 O+ g& R! }
so

6 C2 F4 z* k$ }7 M1 zb=-kc+s

/ }2 @  G6 a: E3 Xfinally:  c= (s-b)/k

3 m7 y) g& F% g$ N3 ]3 J
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:
# n% I* i* l& y. ]4 o/ [: T* Z请教大家一道微分题，多谢！
; i% k6 m* m) F# T) d
d [(a+bx) ] /dx = -kc +s
6 F3 i$ y; b! [/ P# S1 t, wwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
- |, H- h1 ]9 B" a
多谢了！
+ D0 \  V: Z4 p, f  {
" H4 T$ V# p6 A" C# w, S4 j[ Last edited by 醉酒 ...

. ^4 X5 W8 j1 K7 Y6 U  K/ C3 l! osorry, did you post another question? your statement is still not clear.
- Y; o  @7 ^* I# A' G/ }" S: U( G
, N- S- i& `8 Y, S7 ua, b, s, k are  constants? c means c(x) ?
1 H# h) h2 |' T4 r/ K1 c
please 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.

/ U0 h: b7 J1 x8 Fa, b, s, k are  constants? c means c(x) ?

please tell me.

对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s

  Q* A7 w. f. R0 O0 d多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
+ y# W3 A# o( N对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s
  T% B$ ?" [8 ?. ^/ ?* N& q( ?
6 T* R. N) `/ d0 g; L多谢

8 g1 n( m/ y" L/ O: i  o) f9 tdo 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) ?

: @6 e8 G! s1 v+ mif 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:
1 g8 p+ x2 ?+ |' a4 V: t( Xdo you mean it is:

' G  ]- f. m8 \% vd { (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 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:
% t7 i, J- c- h. R( n0 G/ {; U
C(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.

4 X7 Z) V9 s2 p- O+ D/ N
+ g* B; [- d! y* jprocedure:
# n! \, @$ |% ]  L( @7 I4 b# a6 N
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
' G( L7 U( I2 f6 E5 Yso:

bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
( ]; i% A0 e" i9 }/ M/ U. H$ h9 n* b
# E  s- h. N9 n8 y% U5 f" h  ^, S(a+bx) dC(x)/dx  = -(k+b)C(x) +s


introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
6 H2 r4 h& Y: b, j. |which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
3 D1 `+ p" Z0 S* p+ Utherefore:

{(a+bx)/K} dY(x)/dx=Y(x)
' Y* Z% z, [, Z7 r3 q3 v- ]! i' ~$ P
from here, we can get:

dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)

so that:   ln Y(x) =( K/b) ln(a+bx)

this means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
( q( S! w' U( x
-(k+b)C(x)+s = (a+bx)^(k/b+1)
# q1 O6 B5 Y5 {5 x! z/ H" n% @1 y! e- y
finally:

  r$ r" t/ s$ Z4 H4 I* H' ~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:
; n) g8 M, \5 i( z2 P$ I8 v: c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。
9 G, G9 }3 P4 N5 L7 z您说：

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

. a( `& y9 n5 ^  Z9 V$ D' }7 A! B# I/ V. r也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
& H% Z* ?, f. K$ K, K- }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。
, P/ C5 M* _6 ]$ b3 w# A( j$ X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。
因此：
+ d9 P. s, o- Y8 W
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
4 L. L( b' v  ^最右边是x的平方。
* b8 r9 n. `  h' b1 p
所以您解的有问题。

[ 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：

9 F* B; N3 C! u) }9 Y/ E! K/ Md{(a+bx)*C(x)}/dx =-k C(x) + s
! Y% I8 {5 k, u7 T0 b' O, u- L. zso:
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
但这是不正确的
: {" ]1 U% q5 u6 C/ S; Hd{(a+bx)* ...

7 \: R9 n" g: o3 q# G- U' x  hPlease 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
) Z3 _* D, T) ~: Y. H* Vd{(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。
) C, t1 U4 \5 Y
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
....
3 k5 w" P0 w& X9 t7 r$ i" H% Zd{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

  {$ i) I7 T: A* C. I" o
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
( ]! p: m6 B% r8 h6 j" T* g
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 V' o4 V$ E) |8 g1 D1 |[ 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:
2 q3 g" m' U2 j) _1 I这样算混淆了微分和导数的概念。

' k9 r  L7 y: Q( O( j4 A' i% od(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

# Q/ U* c3 I% T/ R4 X' fIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

. |  R8 T* s% g$ ~3 }+ s
3 ]  e  w* A+ a: D! R( b这样算混淆了微分和导数的概念。
3 a' r5 k  T8 v$ x2 X
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
; G) E5 t3 P1 e$ J( z5 ~0 `
1 L! x" t$ E  T- l' nAh, you made a mistake: it should be:
$ T" n# i5 X" R! d. p6 o
9 Y1 d! x  s4 T  t( O8 k0 ^d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx


( C0 N4 d9 }3 s8 e4 k9 P5 _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)
4 w' x5 C) Y- \. H! z# |* t; h
for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

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。

1 e( Z* n( e9 f( vd(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
& g% x+ {7 n8 n) g- ^
Ah, you made a mistake: it should be:

6 Z& g& l0 d; q% r, ed(f(x)g(x)) ...

2 F" Q4 T0 M$ h6 T7 c! |
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.

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.

# @0 @# D/ U( O  ~8 D
! Y3 M+ J: P7 V( G; d# rThanks, 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:
4 D' v% r9 R6 t; d$ v
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。
; c* C) _5 w/ J. F0 `9 B! y, 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 ...

0 t, E" I9 l" X3 V& l
( y% u8 E! v) q4 ~& }( h6 m$ T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？

1 M5 q2 R# z; l* r% c) e# e
/ H6 d7 R4 h6 {; i5 u% M) Q+ Q* 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。