请教大家一道微分题，多谢！(原题贴错了，现纠正）

醉酒当歌

请教大家一道微分题，多谢！(原题贴错了，现纠正）

d [(a+bx)c ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?

* R/ g9 S. n: O0 }  `多谢了！

[ Last edited by 醉酒当歌 on 2005-4-4 at 11:33 AM ]

sindowa

d [(a+bx)c ] /dx = -kc +s

1 p4 v. E/ s! \& G$ o1 `(a+bx)c  = (-kc +s)x
ac+bxc=-kxc+sx
! Q9 Y4 z2 V8 }/ R6 C( v(a+bx+kx)c=sx
' T4 k( t6 K  `c=sx/(a+bx+kx)

十年移民路_

Solution:

From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
  K% a' H$ i: Q  wso:

) f# y1 i# k! v7 h5 n1 f. obC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.
8 P2 p# R# A4 ?5 ?# Y
(a+bx) dC(x)/dx  = -(k+b)C(x) +s
; L6 M  w. O9 V- m$ y
% l9 ]. {) t8 u7 o$ D; s
introduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
3 c/ |  x% i. @' Ctherefore:

6 o4 `0 L; p. w- @5 h" Q/ I9 E{(a+bx)/K} dY(x)/dx=Y(x)
* o! l5 Y% _: `# N9 ]( |
* b0 A7 L% @0 f) y, N' Z1 bfrom here, we can get:
1 y: A- ^: K7 |6 a8 K9 t0 n
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = K/b {(a+bx)}d(a+bx)
; E" w% q1 L, h$ [2 ]; a9 k0 z
so that:   ln Y(x) =( K/b) ln(a+bx)

0 U8 i( @+ f4 Gthis means:  Y(x) = (a+bx)^(K/b)
2 D5 W, R6 g9 B+ a  ^1 O, m. Sby using early transform, we can have:

" o8 w6 N5 K; L1 E, m4 D-(k+b)C(x)+s = (a+bx)^(k/b+1)

# E$ V) z1 J4 {finally:
3 v5 Z4 H- _$ E! w% C' I/ X3 r$ D" A
: Z- ]: k! B. Q0 t1 }  AC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)