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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
老杨团队,追求完美;客户至上,服务到位!
请教大家一道微分题,多谢!(原题贴错了,现纠正)) i2 O7 y6 Z2 q( }( P
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d [(a+bx)c ] /dx = -kc +s ( g* T! I3 K* H3 [  \) G. }: A
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?( N4 S. f' i% K; h3 D
5 P2 \" x: T- G
多谢了!
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[ 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
* t! f, h, `8 K$ T) {* q* A6 Q2 ^: N. V3 U
(a+bx)c  = (-kc +s)x
- G( J; W' {" P' U+ J* ]. O; xac+bxc=-kxc+sx
/ `# Q* @1 M' q; v* Y- J(a+bx+kx)c=sx
% [. G" z* i& `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:
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1 J% V! f9 g- h; {5 }+ DFrom:  d{(a+bx)*C(x)}/dx =-k C(x) + s
- F5 x9 V# Z" f( Y+ \5 ?so:
7 Z, b. \: ?9 d- h2 {% v
* T: a+ n9 d/ |! \  pbC(x) + (a+bx) dC(x)/dx = -kC(x) +s6 r, }/ z4 o& _. Q1 N6 L
i.e.
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(a+bx) dC(x)/dx  = -(k+b)C(x) +s
* h" S2 p6 A5 R5 {9 c: Y1 \; _: }0 K

& C, s! x3 u8 r: E. K$ K0 ~introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
" u% H0 {+ G( s1 B5 Twhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx! B# S/ J1 |3 n" ?' B( U& v- @
therefore:3 v0 A4 z$ E- r: n& O: Q2 s- h9 ]0 ^1 z

! w4 \8 r0 w) i4 ]" Q9 l{(a+bx)/K} dY(x)/dx=Y(x)
6 g+ d  Y. q3 J7 @/ \7 W9 j9 N
$ I" E/ G( o& J5 t  Bfrom here, we can get:  U0 U! }# b( L- W$ K, k' ?

7 }6 U2 y5 r  S9 x/ ~dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
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so that:   ln Y(x) =( K/b) ln(a+bx)
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this means:  Y(x) = (a+bx)^(K/b)/ \4 `3 L/ u5 |* e& t- e
by using early transform, we can have:- n+ G* a  ?; |2 K$ t
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-(k+b)C(x)+s = (a+bx)^(k/b+1)7 H) f+ G% N5 U8 c

3 [6 z! E% D% i) pfinally:
0 \+ n" A; x( |" T8 Q9 E  J2 C; N) o# O2 ?; d
C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)
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