 鲜花( 19)  鸡蛋( 0)
|
Solution:
8 \! ]- T) s1 ]+ c) `) b! T& U( k" t0 U
From: d{(a+bx)*C(x)}/dx =-k C(x) + s( Y' x% g& a$ P/ ]" Q7 J9 f8 M
so:2 |" X/ N, j) C! g/ o
; I& Z" V, ~0 }* N: J
bC(x) + (a+bx) dC(x)/dx = -kC(x) +s
; W) f- O' {, D; L" t' W9 `) qi.e.
! o( E1 ?1 \' e: y; v3 f" n; ?" }6 z- _9 Q
(a+bx) dC(x)/dx = -(k+b)C(x) +s8 E: }- z# d B6 R; _
$ I- m8 E; ~; v3 X% U u5 t
5 m' |# U+ e! U# Q1 M" K/ @
introduce a tranform: KC(x)+s =Y(x), where K=-(k+b) : t$ N4 l' A0 a8 y% \0 P
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
3 e6 O* g$ A0 l m% P2 r" etherefore:
% `5 t$ b$ @5 `' u! }: T
5 x, c3 k( G+ }6 t4 _3 \3 r{(a+bx)/K} dY(x)/dx=Y(x)- w6 i4 ?1 C6 H1 R' K; J1 W9 K/ h$ E: z
1 L [/ Y. b5 v4 t5 g1 U1 F& H
from here, we can get:: M6 J( k' L0 u, w) R' G* D3 `& K
1 `! s8 J% i- |8 G9 ~- EdY(x)/Y(x) = [K/(a+bx)]dx i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
8 ^/ ~- B) F+ o7 ]1 X! c
( U0 R& {1 B# w& u' x. Q, Fso that: ln Y(x) =( K/b) ln(a+bx): i* t! j, H; n- F5 C1 G2 i
8 C! y' G H/ T* G0 _1 B' }
this means: Y(x) = (a+bx)^(K/b)1 w- `4 D) W4 B8 v
by using early transform, we can have:
V- I, X+ o/ b0 v0 |" l; W8 T
/ F; Z4 N6 g; N7 v-(k+b)C(x)+s = (a+bx)^(k/b+1)
8 Z4 t( e, e' q7 B6 }! L9 }
6 s) v5 M" F; y% L! A9 M& sfinally:# p; I7 T% W9 p/ V4 e
5 P, D( q& A0 @( H; D. p
C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s) |
|