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）

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?

( H" L! f% g: G5 X7 s* ]0 i多谢了！
3 o) v+ k7 X# x: ^
[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
+ Q. D! K' v" Y7 ]! E, l+ W
(a+bx)c  = (-kc +s)x
/ D: G& |# \; k8 W) g4 \! `ac+bxc=-kxc+sx
(a+bx+kx)c=sx
1 ^8 ?% C/ O! x  d4 W: h: ^c=sx/(a+bx+kx)

十年移民路_

Solution:
" A  m$ H! m2 }1 y- V
8 A0 |! K( V7 {. q0 lFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
  Y% O& m) x$ t3 N! ?, {# k" Qso:
9 E6 h3 \: R4 |7 L" e, y! [% H
! f, Q! E" `+ }. obC(x) + (a+bx) dC(x)/dx = -kC(x) +s
& F0 K& ~0 B1 _' ^i.e.

(a+bx) dC(x)/dx  = -(k+b)C(x) +s


0 {; |7 C- ]4 kintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
. R# y3 E- o8 cwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
& Z* ?, c! g$ W- F( Ytherefore:
0 [0 {6 @$ }% t
{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:

' L/ Z/ U9 Y* t3 k0 _/ ~1 ~dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)

4 D; |3 P! A; f9 ?8 qso that:   ln Y(x) =( K/b) ln(a+bx)

this means:  Y(x) = (a+bx)^(K/b)
1 i1 l2 ?3 [1 D! P5 rby using early transform, we can have:
' g; E- v' v' U
% b' T& O8 b5 W. m4 }0 e- c-(k+b)C(x)+s = (a+bx)^(k/b+1)
# n/ V$ v: Q0 A- w# N
' b. j1 s" B5 [4 Dfinally:
( C0 ]3 X8 k; N$ D7 C  G: }& f/ p
C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)