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
# C9 e$ a7 T+ r1 X1 O! c+ qwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！
0 u# Y, [: N$ D9 V* p- q$ \
[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

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

( }( \% ^4 H8 o. q$ p(a+bx)c  = (-kc +s)x
+ l, f" E# t6 y: U! U* L. Aac+bxc=-kxc+sx
(a+bx+kx)c=sx
. j# p' D; f- Y  Pc=sx/(a+bx+kx)

十年移民路_

Solution:

6 o6 q8 Z" U" \From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:

bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
6 }  H. e# [; U! Si.e.
( m' ~8 \9 W" D/ K) i5 W2 E: ~( K
(a+bx) dC(x)/dx  = -(k+b)C(x) +s

$ l8 ]( O6 u: z5 x
' O! x+ w/ U* ?( h- Yintroduce 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
therefore:
7 S9 E- t# ]- e6 X
{(a+bx)/K} dY(x)/dx=Y(x)
3 D" F2 I8 y$ D* m
. D, y  M3 }4 J& Q9 `from here, we can get:

dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
& \1 ?5 b; d% _. C
so that:   ln Y(x) =( K/b) ln(a+bx)
6 B7 L5 _5 _% `6 c
this means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
; @7 T9 G- n% |1 g7 J$ g: h1 |
- |+ N5 `; L2 c# f$ ^( |& s. n-(k+b)C(x)+s = (a+bx)^(k/b+1)
; h) t/ }4 ~% g% e4 @( Y
finally:

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