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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)2 L$ ~7 f3 I" Z' i3 H! {/ ^- S
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d [(a+bx)c ] /dx = -kc +s
3 A$ P4 J. G8 r, o1 ywhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?, l& }9 c& D0 m9 _( n
5 t9 Y$ n5 w/ d: ?8 N( B2 D: z
多谢了!/ v4 e3 L6 x  f3 q* U- l9 w6 f
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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
2 O+ _: D* i2 S8 H7 u) k. J4 E6 @2 O) I& @* l( z" @7 z! `0 W
(a+bx)c  = (-kc +s)x
( w  D! r1 C* J2 F6 ]' _. Rac+bxc=-kxc+sx+ X) X. S  k; |7 `& O
(a+bx+kx)c=sx
6 ~$ \" d& C  r- W* @$ v$ ~9 Bc=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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6 Z/ o& g/ Y+ C, L5 S7 ~From:  d{(a+bx)*C(x)}/dx =-k C(x) + s1 i3 z' m: j, N0 _% _
so:! ]7 a! A0 [* `* |

- V- ?) K$ U. S: qbC(x) + (a+bx) dC(x)/dx = -kC(x) +s/ k3 w& O  P9 m$ [$ v# Q* e
i.e.
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9 z3 o. x  a' g5 i& W(a+bx) dC(x)/dx  = -(k+b)C(x) +s. k% w: ~- N& A3 Z

: L: q: \7 @$ r
$ u! v) s& Y2 j+ n1 n- }9 z3 l' Ointroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
; `+ i. h2 ?. E  u0 N. I$ F& ^which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx+ p9 w2 l' `. S- r3 O9 x
therefore:( X- H* H, ~# ?5 Y8 u6 N0 N

+ T3 y5 _( p6 K+ p{(a+bx)/K} dY(x)/dx=Y(x)
9 h1 q3 P! K1 q% ~. ?3 k/ \( ^4 z# r
from here, we can get:
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dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
: D$ f0 r2 E$ m2 V/ B# o" O$ R3 b2 l: L* p, Y
so that:   ln Y(x) =( K/b) ln(a+bx)
, S0 N/ |+ }$ {8 }
0 _$ u! p! V) K3 ythis means:  Y(x) = (a+bx)^(K/b)1 x/ A: ^3 L9 @( O  P
by using early transform, we can have:5 [( K9 i: ]1 y& |6 D* \
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-(k+b)C(x)+s = (a+bx)^(k/b+1)" a+ G% ^' ~; ?1 L6 z

. ~$ ]- r0 a" c' E: kfinally:
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  j* ^3 ^6 N( E/ J; Z) t$ P$ o: `C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)
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