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）

3 w4 j% i* i- N* Pd [(a+bx)c ] /dx = -kc +s
8 I+ t9 E: G: d" A: f! Nwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

多谢了！

* a: \: {; P. F0 c7 s! Y6 U  F[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s
3 N( b) }* g1 D5 i
+ Z% g) O8 |( u+ E6 ~2 B$ r(a+bx)c  = (-kc +s)x
* B6 V. H$ x2 c9 W; Q# u& q4 ^ac+bxc=-kxc+sx
(a+bx+kx)c=sx
c=sx/(a+bx+kx)

十年移民路_

Solution:

8 b7 ?5 W+ i6 l: ?From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
2 W4 z0 i9 ]1 P7 I3 x$ O1 s/ s
  J: B+ q$ p& u5 W6 `- [bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
' O/ J! \) R9 }1 \* ki.e.

(a+bx) dC(x)/dx  = -(k+b)C(x) +s
( ]: q0 ]9 n$ O  @
- R& b1 n8 o: I0 k& }
" i! c6 u  z1 d8 [0 [introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
( B) Q( y3 L, s, P& ~" owhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
therefore:
( U' `8 z8 ?4 F9 b# p, R) |8 K
{(a+bx)/K} dY(x)/dx=Y(x)
) J. ^( E. d8 n$ b
from here, we can get:

2 W" F% G$ c" i6 w1 O" d( X  D" ldY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
' J7 X1 e2 X4 ]$ s
0 @0 X/ p7 a& ^& [2 Nso that:   ln Y(x) =( K/b) ln(a+bx)

this means:  Y(x) = (a+bx)^(K/b)
% L$ Z" c9 f" Z% q+ x5 ?" }by using early transform, we can have:
* @, R: m" v( b( E
-(k+b)C(x)+s = (a+bx)^(k/b+1)
; u- l( {, t% s/ C" v
8 \9 u! U+ T  l4 A! S& A$ n1 lfinally:
# H) J& x+ ?' m! t: z. K
C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)