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?

1 Z( O. y$ q# W; k" p5 m8 D多谢了！

[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
& i2 v1 P/ P7 y/ v8 C) }' G
(a+bx)c  = (-kc +s)x
2 E  e6 w" ^) v9 j- r9 B& Iac+bxc=-kxc+sx
(a+bx+kx)c=sx
! }' q4 G4 t0 [3 H# ?6 f; N$ q' |c=sx/(a+bx+kx)

十年移民路_

Solution:
% b- j* \) g  s8 G3 g
; H3 U! x0 T+ u0 |8 f) JFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:

0 _4 t/ L7 S* MbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
0 s! s% I7 c. P  Y+ q, u
(a+bx) dC(x)/dx  = -(k+b)C(x) +s
0 ~0 F0 C4 f/ L1 `6 H

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
* }% @4 S( H2 vtherefore:
5 y# A8 s3 l3 {0 X$ v9 X
" Y) t: J" a# A& d: l8 {6 b5 J8 y. L{(a+bx)/K} dY(x)/dx=Y(x)
  h: b! R: E8 i* P3 X, j; r  a7 Y
from here, we can get:
2 A3 s2 L" z/ x1 Y
8 D; ]+ G7 b+ u$ vdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
3 E2 S6 u0 w" F2 X$ B5 D1 T
so that:   ln Y(x) =( K/b) ln(a+bx)

this means:  Y(x) = (a+bx)^(K/b)
7 B1 ~: d  x( O- H7 h: E3 ^by using early transform, we can have:

-(k+b)C(x)+s = (a+bx)^(k/b+1)

finally:

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