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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)& z7 C" o( Y6 e

6 a( Q/ s+ m0 q% J0 Fd [(a+bx)c ] /dx = -kc +s
  H# ^  W6 W( B6 ]3 d- I. E* E5 Swhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
% h& |; p/ c: a' C3 b, P
8 P! o# j( E( q2 R多谢了!
! g# A' Z. \- u6 {6 M: y! {$ M# ^+ T$ z5 M' w, J, u: ~9 E
[ 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 +s, j8 Y7 O8 f6 U9 L
, s3 \* i& @3 ]( c* G8 l7 K/ @# i
(a+bx)c  = (-kc +s)x) j" |; ?$ T' o# g) I( |2 Y5 u
ac+bxc=-kxc+sx
5 |& V+ T7 S; ~* i(a+bx+kx)c=sx
7 j& s0 j/ B; d9 r* y6 G# D! D3 ic=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:7 o7 _  _9 ~: B9 G- i$ s6 g
: h# T: [0 P, S' ^5 j7 b
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
1 W8 B7 t  u: {& X) ^so:2 v. ?" n) P# p" u  I
8 m  m- B- Y1 R; W1 J! l
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s3 i! u7 G; e- s5 X
i.e.
) L3 F; G, r# B! Z
/ f# {  Z& o% ~/ ~! _5 ?0 T(a+bx) dC(x)/dx  = -(k+b)C(x) +s% o2 Z  n2 V* M" O

8 D" `% ]8 {! M4 ?  S9 a9 S) q, g
& G5 B# v; R2 P, ^7 {) M5 Jintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
/ v! K' d& L: }0 o1 `: Vwhich means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx! W. G6 v& \) ?0 J. q3 L
therefore:" v* l" v1 R+ j2 k: k$ D5 U5 Z

/ Q5 V  j' ?/ C5 F8 _8 q{(a+bx)/K} dY(x)/dx=Y(x)
2 R7 U5 |" y/ q  L
4 K# Y0 I( F5 U4 E: P& ?' u) Mfrom here, we can get:; r  A- ?. w& x: t5 J- a  c
& }1 r8 r1 \( E2 K3 w; e9 ~( _
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
8 @7 D% `* N: I( B2 ?( y2 m
4 {3 Z9 P1 G  K! nso that:   ln Y(x) =( K/b) ln(a+bx)* G* U6 G0 @+ |# [* ~6 Y2 N
1 g; s( Q6 E4 ?) F. M  [: |0 F! G- e
this means:  Y(x) = (a+bx)^(K/b). ^' }$ }$ c7 u
by using early transform, we can have:. k% e' D, w) r+ ~8 j

5 Y: f% r9 W) Q* q- I-(k+b)C(x)+s = (a+bx)^(k/b+1). k8 _" u4 h* X

% h# o# P+ n0 [* Vfinally:- m' Y" C; R$ G1 ?  {  I/ K

4 l# U( M0 p6 q/ C/ AC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)
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