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- X: H/ I0 q  \1 K; L1 a, R5 S  B7 cd [(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?
2 l! C$ r/ Y. E- L5 X7 B+ e
8 P1 K  Y* r1 w: e6 @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
ac+bxc=-kxc+sx
(a+bx+kx)c=sx
9 _" B+ Y4 C+ v& h. |/ hc=sx/(a+bx+kx)

十年移民路_

Solution:

# f6 d( Y2 [9 q/ U: W" l% V, J. F+ EFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
$ t0 l! G, Q- Y6 k2 z
- p# J. w4 C& d0 m+ M& s0 PbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
; u6 q2 Y6 N% a% ^, m7 H: yi.e.
! j8 K8 f( l3 ?' f/ @3 [
(a+bx) dC(x)/dx  = -(k+b)C(x) +s

; D0 S# H5 Z% o8 j; a, _! D
5 k9 ?$ S* s8 {introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
. [# G+ @1 w% T  |2 l6 pwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
therefore:
5 E: ?! [; \' c) ]$ K# L
$ z( h1 l  Y, P8 U3 |8 _. q{(a+bx)/K} dY(x)/dx=Y(x)

: X. \  G+ M/ s. D. tfrom here, we can get:
1 f" ^5 o: n. D" K6 [* _% A& g
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
. w! g& p1 {# A4 I
so that:   ln Y(x) =( K/b) ln(a+bx)
/ Y; R2 B4 U' c* D4 @9 G
this means:  Y(x) = (a+bx)^(K/b)
" |7 @. T$ h- D4 S: Hby using early transform, we can have:
$ X1 `& L5 _! r" Z& D
; r4 _! r4 B6 ]% O/ x7 C-(k+b)C(x)+s = (a+bx)^(k/b+1)
1 F0 y4 y  s- B+ |% K8 u
finally:
" k. i. o) I$ T) |3 t) b  y- K3 U
C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)