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
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
5 s& f" Q9 n* a3 l
多谢了！
7 F9 J+ ^' m; Q5 h! _# ^
[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

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

7 k9 X; u/ o% Z9 Q(a+bx)c  = (-kc +s)x
ac+bxc=-kxc+sx
(a+bx+kx)c=sx
c=sx/(a+bx+kx)

十年移民路_

Solution:

. p5 k, P- I* o9 [From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
) a5 ^% Y' M  aso:
% {: g$ T  P3 L" U' o; g6 F
$ X1 S+ T- `, ~# y5 _bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.

(a+bx) dC(x)/dx  = -(k+b)C(x) +s
1 U1 A- k. a; i

introduce 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
1 k1 i% E( [* g' L4 n  _3 W/ Htherefore:
+ k# G1 z( b2 H+ J! L% A0 x) i
* }! q1 U1 @7 Z* U/ }4 H{(a+bx)/K} dY(x)/dx=Y(x)

  Q* C) X( A, {0 Q( O7 K4 l) g* Dfrom 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)
- r3 U, Q8 _+ W( K
so that:   ln Y(x) =( K/b) ln(a+bx)
% @  e% K/ Q/ Z# Q2 _/ o  z
this means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
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( F% I' }' b, `# \& ~-(k+b)C(x)+s = (a+bx)^(k/b+1)
+ Z; t- }9 Y) b' f
finally:

+ h+ ^7 s- F8 W/ v* {) b. \C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)