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）
* q/ m  f, H9 }4 n3 W5 l! v! |
d [(a+bx)c ] /dx = -kc +s
% I- f) i% G* e' e* Z2 s$ ~where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！

; u  @+ k# i$ U" ^8 i* h: E[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s

(a+bx)c  = (-kc +s)x
7 `1 S6 E2 g/ V! E4 }' Vac+bxc=-kxc+sx
(a+bx+kx)c=sx
  s% m) c) n* ^c=sx/(a+bx+kx)

十年移民路_

Solution:

0 [" a$ G/ j3 @# j4 aFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
+ \& d! C- L- ^- nso:

bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
' ~0 j2 A9 T( y/ Q7 Fi.e.

% @! N; |: A& B(a+bx) dC(x)/dx  = -(k+b)C(x) +s


+ K$ B, w) m+ O2 Xintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
) K  J2 f# A* p' c1 N, Zwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
  F2 `+ n7 p8 |" \) w% Ptherefore:
* w  M$ a, I6 J4 H3 w
{(a+bx)/K} dY(x)/dx=Y(x)
) n( K; j( e+ y
/ w  S; h$ _4 Q, Z/ B/ G) L8 Lfrom here, we can get:
9 [5 X8 i6 N; F6 s
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)

1 l7 `6 Q4 A1 g* }# dso that:   ln Y(x) =( K/b) ln(a+bx)
* v0 ]# W$ |" y) h* t) V
this means:  Y(x) = (a+bx)^(K/b)
1 y* C) j' O. n6 Aby using early transform, we can have:

-(k+b)C(x)+s = (a+bx)^(k/b+1)
+ X' x5 a7 O( @" o
finally:
. m$ j6 [/ L; T& k& p6 e
( d' I8 a5 t* h" w- u" oC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)