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）
' F! W  F) o$ j8 j
d [(a+bx)c ] /dx = -kc +s
1 W' X; a$ V& Z# b* U( _. _' R4 Cwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

* F! k8 [# v% ?3 u6 ?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

(a+bx)c  = (-kc +s)x
2 f6 ~1 O& X$ W! Q, t/ tac+bxc=-kxc+sx
" F) V( h; G* B4 T1 ^* X: J. \' w+ Y(a+bx+kx)c=sx
c=sx/(a+bx+kx)

十年移民路_

Solution:

From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
& N* k% E8 C1 \: S
- d+ L2 C7 A2 X6 KbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
- d$ Y5 |0 ^% E9 r* z) Xi.e.
% {6 _0 G- ^) g, h" l
: y: R* Y& z  k(a+bx) dC(x)/dx  = -(k+b)C(x) +s
! Z" U0 E/ E; B* T, h
6 K; P$ j$ C5 w
7 c/ H- [8 @' B8 N# t! N  U0 Pintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
; w8 [* R6 E" f2 |4 \; l  gwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
; N7 Q! e+ V) k- E* Jtherefore:

{(a+bx)/K} dY(x)/dx=Y(x)

from here, we can get:
8 \" A) O- Z2 ~! @+ `
0 N4 O9 M- u  H8 X8 n+ TdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
2 P% M0 B4 ~9 G) q5 F/ Z
so that:   ln Y(x) =( K/b) ln(a+bx)
$ v( }7 p  k: U8 d! V" z$ V7 ~
2 P- A* |6 n5 [this means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
& V7 f' T) a% t5 H- e- X% \
% m, f3 l5 {2 ^-(k+b)C(x)+s = (a+bx)^(k/b+1)
0 j+ o9 c+ ~- X* P$ N. Q
finally:

) v0 J1 _- l+ P: K: cC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)