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请教大家一道微分题,多谢!(原题贴错了,现纠正)

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发表于 2005-4-3 19:44 | 显示全部楼层 |阅读模式
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请教大家一道微分题,多谢!(原题贴错了,现纠正)9 [6 F( P1 y: B  L1 m* _, {

, L$ g. g3 J+ _d [(a+bx)c ] /dx = -kc +s
. c2 B8 w/ p; f) s4 m5 owhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
; X; G0 T; u. [1 w* Z
( ^  U6 G! g" ^4 L4 E4 A' z多谢了!- F7 n4 u% F* x7 q6 Z
3 O$ ?( ^) I  m  i
[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]
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发表于 2005-4-5 02:47 | 显示全部楼层

供参考

d [(a+bx)c ] /dx = -kc +s8 t0 @% y, R+ g
* L/ L% T) }& J  Q" e
(a+bx)c  = (-kc +s)x
6 N* d3 Q. V6 C  o. t8 j( eac+bxc=-kxc+sx
; \; [7 Y% A" |) V(a+bx+kx)c=sx
* @5 E, i* S4 b# {3 Qc=sx/(a+bx+kx)
鲜花(19) 鸡蛋(0)
发表于 2005-4-5 22:11 | 显示全部楼层
Solution:  {& H$ l* ~4 k% _

: A( n& \- H7 sFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
9 l4 X3 u! J5 C( Uso:$ f+ @( F/ K6 ~+ h; s8 P
# L- g* K) _3 M9 e( T
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s* P' D$ v) N. c
i.e.
  S# Y7 W# a6 v2 x( R7 N/ F/ G3 f2 L! Y( M  L
(a+bx) dC(x)/dx  = -(k+b)C(x) +s2 G% X. c) R. H) w/ S. ?" _

" L) `5 [0 w5 T6 S5 D2 a/ u, n  _- N  u: _" K) P1 b
introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
6 l$ C/ Z5 r2 Q( ?! B3 a4 ~+ e5 @  hwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx" A- D" {& J. o# ~+ h4 K  k: S0 a
therefore:( m' ~7 U6 y/ o8 w, l" W

- f" |5 ?0 f$ R2 Q0 F: G5 ^/ [1 f" W{(a+bx)/K} dY(x)/dx=Y(x)
( y7 m6 c2 I/ ^$ C. _( ?5 c
" a: E' ^  }: X. Wfrom here, we can get:7 i3 q, o/ Z" G5 i

7 ?4 I5 g. t( x( V6 y0 N- U& KdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)% {. O+ }# `. `  n% i6 i3 F) p
' T8 b$ I1 t0 j$ b2 t
so that:   ln Y(x) =( K/b) ln(a+bx)
6 D! ^) Z! |' A3 C. x- p0 X6 L5 }) l. Z( j
this means:  Y(x) = (a+bx)^(K/b)$ @; v7 q! l1 i2 ?8 Y
by using early transform, we can have:
) m% f) W1 S  }& s2 |, B# a
" b! ?, e9 ^- Y% n, V-(k+b)C(x)+s = (a+bx)^(k/b+1)5 w: y3 g4 U5 y1 t+ i/ ^' _3 U- T1 L
# _) H" Z5 ~: c3 \6 Q+ S9 b! S
finally:
" G- `$ Q" |4 s: L  T% ^0 e
) }% B% R9 X/ Q+ k8 LC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)
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