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）
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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?

/ Y6 a1 e# O& r多谢了！
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+ B' [- I/ `# |5 h[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

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

% j5 E5 t- C) m! N4 T$ `(a+bx)c  = (-kc +s)x
ac+bxc=-kxc+sx
(a+bx+kx)c=sx
1 Z  `  }6 f! w& U4 J) J) ic=sx/(a+bx+kx)

十年移民路_

Solution:

3 U; a, h% T: [, ZFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
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! r3 f: `8 l3 D( F% V6 VbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
" {+ L2 \- C+ i) T# Y0 J2 ei.e.
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* V8 `# _! C) x+ i(a+bx) dC(x)/dx  = -(k+b)C(x) +s


" h& O6 `* d# N0 V) Uintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
# e: T2 z/ Q; k7 d8 [/ Rwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
& ^' R: x* J; x0 x4 Rtherefore:
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5 `3 C, V+ h  W  \! I* X% c{(a+bx)/K} dY(x)/dx=Y(x)
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from here, we can get:

, s5 ^$ |2 T7 ]  n7 UdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
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4 X2 n+ u0 _& q) W" E9 Sso that:   ln Y(x) =( K/b) ln(a+bx)
3 t) O6 j+ u# ]7 ?, K( L
, m. w3 i, x. Gthis means:  Y(x) = (a+bx)^(K/b)
& L: X: A* w7 a3 Cby using early transform, we can have:
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, n4 h. Q! O0 m2 G6 Q& f# D-(k+b)C(x)+s = (a+bx)^(k/b+1)
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finally:
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3 \# x5 r) f* ?4 X. Q, D- vC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)