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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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鲜花(150) 鸡蛋(3)
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)
' M. j# _6 d% K% p: {& Y; Z. P; N3 \) X( o. F& k. \9 M2 [
d [(a+bx)c ] /dx = -kc +s
0 s9 l+ L* L/ @% O1 v4 m9 hwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
7 ~+ I$ Z4 w- f% s& Y5 K" c4 N. m9 o+ J- `' G2 V
多谢了!
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3 W; k  X$ y" Z[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]
鲜花(0) 鸡蛋(0)
发表于 2005-4-5 02:47 | 显示全部楼层

供参考

d [(a+bx)c ] /dx = -kc +s
  ], J/ c" k0 o# w6 r: J. F% R
5 h9 a( s& R7 v  X$ E; D(a+bx)c  = (-kc +s)x2 O  Y3 W7 `* f. I0 A( B* U% ^
ac+bxc=-kxc+sx. E" Q7 i1 W" z  q( Z) _; \& f' Q! w
(a+bx+kx)c=sx  I4 @+ D2 Y) F8 E6 Y) m
c=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:
' B: Z& M8 w' H3 d' }0 x7 |0 k, r3 d5 H9 P- b& Z4 X4 M
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
3 h. P, \5 C# U6 v' S3 n' \4 oso:0 b# i5 r% E  {( Z* M1 O
1 S2 C) p* W6 H$ @
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s. c) y; E% I0 x
i.e.
+ G( ]& ]) }  o( J; }% I9 ~3 b) k
(a+bx) dC(x)/dx  = -(k+b)C(x) +s3 L' g8 S6 ?: h- Z
* p3 i" I# ^3 D' R

1 H9 G6 `6 `! ?) V8 @, Y/ Tintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
+ _' _3 v% `) k  f( owhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx- k3 s' N4 ]0 Z. s
therefore:
' m* |( j0 @3 l/ {* X
! \8 _* S$ T1 K{(a+bx)/K} dY(x)/dx=Y(x)  \0 `2 @2 A2 U; |
: I8 m8 W* F7 N
from here, we can get:
/ T9 q# ]2 R# R& g( e( J0 o
$ c# w4 Q# k! q$ [. hdY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
$ n4 y0 f: G; u0 ]( X* F3 |% ]- I( K! E3 l
so that:   ln Y(x) =( K/b) ln(a+bx)
  B7 b2 }) f+ Z  F% Q
1 L4 u3 `$ x6 qthis means:  Y(x) = (a+bx)^(K/b)% k* P: h4 G, r' [
by using early transform, we can have:
2 U' b' S/ ^* A0 g2 x( ~8 a. h$ b2 ~" _8 I, g- B
-(k+b)C(x)+s = (a+bx)^(k/b+1)8 h! R8 a  g% b  A% a
5 B2 f7 c! y0 q, |. A$ G
finally:
9 N& w& [( T2 B9 O
- D" z& Y, q4 O3 g$ p& a! O8 bC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)
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