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）
4 g. X1 a1 I" e* M* m! h, k7 B  `
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?

, w; P3 H' S$ q! Z; I多谢了！
% z6 v) G. @7 U: B9 k) w
[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
* s8 B; k3 K- Q( o( l/ r" m
(a+bx)c  = (-kc +s)x
( x" l0 L: l) x" [! i1 iac+bxc=-kxc+sx
2 P7 T3 X' T! l! y5 j(a+bx+kx)c=sx
c=sx/(a+bx+kx)

十年移民路_

Solution:

1 p! H2 l% q* j( PFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:

& [; Y/ T! J* d: IbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
  v# ]* M  N4 K+ C4 T
0 i: y- M* p3 O1 n( c(a+bx) dC(x)/dx  = -(k+b)C(x) +s
# q3 T+ b# H6 a! U8 {3 M0 z% L6 H
/ X3 @% `( o8 e8 y; x+ l
introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
& ^7 x7 X5 e% O8 a0 ^which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
therefore:

% r4 {0 w0 d. ]8 \: [{(a+bx)/K} dY(x)/dx=Y(x)
. ?/ N7 }; p/ i! w
from here, we can get:

" d7 X4 S. Y: d5 F. s( x* I; \dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
8 [* F7 V' u' V8 W1 S
, J) r0 Y& B# ]# jso that:   ln Y(x) =( K/b) ln(a+bx)

this means:  Y(x) = (a+bx)^(K/b)
1 I, u( y9 i4 Xby using early transform, we can have:
. R4 O( x  X' r4 g
-(k+b)C(x)+s = (a+bx)^(k/b+1)
* M, i# o+ h" p. d9 J
$ N3 G, x) T3 F% ffinally:

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