请教大家一道微分题，多谢！(原题贴错了，现纠正）

醉酒当歌

请教大家一道微分题，多谢！(原题贴错了，现纠正）

9 ~% O" d, v" i( A% \d [(a+bx)c ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
. T1 w+ Q& ~; `- w
多谢了！
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2 H# \' P8 x- H* B[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
, S3 x3 M5 W' T. \) T) c- m
; Y  r+ b  R2 D; d0 P(a+bx)c  = (-kc +s)x
9 M% C8 W+ X1 `6 Y; `8 X" N  l( bac+bxc=-kxc+sx
(a+bx+kx)c=sx
c=sx/(a+bx+kx)

十年移民路_

Solution:
: S3 g# R" l( X: j/ `0 F& w) p
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
) ~/ T8 u& K. ~& P, nso:
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# D- E7 e6 t. w. [bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
! {) v. y9 w& b/ Wi.e.
6 `# o. C" ]" M9 H6 ?& B9 \+ r
(a+bx) dC(x)/dx  = -(k+b)C(x) +s
1 S3 f4 L1 ]" w: y) p7 M: J. N

introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
; f% r! x7 x6 ?which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
( z8 D6 v! T: Z# a9 vtherefore:

( ~7 F! ?! _7 @) G6 B8 ~: l! P{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:

+ c3 B1 i5 A9 v1 kdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
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" c$ b! k7 k+ Z9 P6 E8 d4 ~7 {so that:   ln Y(x) =( K/b) ln(a+bx)

, X) L  Y- N8 I: u- v9 Hthis means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:

-(k+b)C(x)+s = (a+bx)^(k/b+1)
9 x7 ]8 f5 O, p( T8 l1 r4 D+ g
  F1 B$ g( ~( Wfinally:

C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)