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）
5 X) O2 F! H' |! Y; A& O* _
d [(a+bx)c ] /dx = -kc +s
5 n) G0 c/ I# c% xwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

" X4 [- U3 ~- ]9 R多谢了！

5 S; c0 r: y& L2 b2 H  h1 z6 c  c[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
1 s6 F! ?2 b- t3 f; ~3 c3 I
# y2 u8 {, d# j0 {+ s+ v* s' o(a+bx)c  = (-kc +s)x
ac+bxc=-kxc+sx
# n# p5 e- R& n/ ?$ o(a+bx+kx)c=sx
, f4 h( }1 L/ I/ {c=sx/(a+bx+kx)

十年移民路_

Solution:

From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
  C$ ?  X0 H& ^8 M  C3 U1 nso:
- N5 k. X3 ?  f! h1 ]3 Q! Z5 h; A
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
  c3 v* H6 l) T) {7 @/ D4 ?1 j+ fi.e.

(a+bx) dC(x)/dx  = -(k+b)C(x) +s

1 V0 X9 o: a( V$ u2 K
. K% f$ t9 Q9 {( @& J, |introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
" h/ E" L9 ^1 dwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
therefore:

  G# [: M# I' }3 W+ x- e& w{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:
8 ^* v4 p: B/ |3 z0 E+ ]+ _
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
) n/ ~7 X* L5 A; R: p% S* E3 y& K
so that:   ln Y(x) =( K/b) ln(a+bx)
3 ^1 m  P% O6 S8 p1 t. M" S
2 J& y" B+ k: x9 @$ @+ ithis means:  Y(x) = (a+bx)^(K/b)
/ d5 E/ G. @2 ?by using early transform, we can have:

1 o2 s" _; f4 z" `-(k+b)C(x)+s = (a+bx)^(k/b+1)

finally:
. t4 T0 j' m9 E% m% V+ l* t# K, [
C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)