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）
! M3 l0 _  G8 P( ^: W, \3 E3 a
" \0 _, d+ @# U6 ^' R3 {9 t& Kd [(a+bx)c ] /dx = -kc +s
- a: B; {" I4 h2 \: G0 ywhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！
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' F4 z3 M9 Y: W$ m, ^[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
+ X( v" M( ~8 H* A5 f
(a+bx)c  = (-kc +s)x
2 L* L9 b6 B& R! w# [& zac+bxc=-kxc+sx
(a+bx+kx)c=sx
c=sx/(a+bx+kx)

十年移民路_

Solution:

; p) ~: V: S+ B  z. C4 {2 ^From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
5 k3 f* Q% D" o4 X* |so:

9 K1 ]" H  {9 e. ObC(x) + (a+bx) dC(x)/dx = -kC(x) +s
) C8 H# W1 _4 l, H. I9 L/ g5 ai.e.
1 a. p, U$ D7 @2 f
(a+bx) dC(x)/dx  = -(k+b)C(x) +s
7 z* w5 K# ?+ H6 h5 p
! {" Q* G9 t2 C5 p
introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
$ T* L! v. r3 ^which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
- e  n+ k6 p: Y2 ntherefore:
& ^: }% ]- h* M" ?+ W: }
  X5 |7 d: b" S+ D/ t{(a+bx)/K} dY(x)/dx=Y(x)
% t( G, N' L: [8 |5 G
8 p8 f" D4 _, V$ g4 T5 hfrom here, we can get:

7 f- A6 p2 f2 _, Y# A" o! @7 @dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
' w) Y9 Z3 j# K: p
- ]# l. P/ d) ]3 J) `7 g) j8 d" ^$ Iso that:   ln Y(x) =( K/b) ln(a+bx)
7 y- x) D: J( P# O9 t( v
this means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
5 b- _* l  [! f* @
9 v7 n- u% J) _: z" @: g$ ~$ C-(k+b)C(x)+s = (a+bx)^(k/b+1)
- Z8 l; [% T! I2 A2 E. s( n
1 F5 p/ r- N- j1 v9 nfinally:

! Y5 b' `* t; f: T' ]) r) NC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)