数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！

d [(a+bx) ] /dx = -kc +s
1 Q7 _0 ?/ ?. n! t6 [7 Xwhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
7 x# k+ C' L) U$ f) W
多谢了！
- o  @8 }. k+ N, K) A- b& ?
" n/ d% p+ |" Y9 G4 U/ X, t[ Last edited by 醉酒当歌 on 2005-4-3 at 07:56 PM ]

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-3 20:41:
请教大家一道微分题，多谢！

+ c; G/ T# t3 F* X! B$ d" Kd [(a+bx) ] /dx = -kc +s
9 h# d2 R$ c2 n, i) P7 J2 r8 l& Cwhere: only x and c are unknown, others are all known,  requiire c = ?
, @- \/ v$ P0 e& t7 m6 F0 Y* O! Q
多谢了！

- D# B4 c; }  q8 _& p
+ G1 r: O) q6 ]" v" Iit is too easy:

d(a+bx)/dx = b
3 @7 b1 `3 I# l0 R, \/ m$ w
so

b=-kc+s

finally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
. ^* c- H8 Z) v4 mit is too easy:

5 u3 T- [$ y5 r& Z- @d(a+bx)/dx = b

so
7 {* t5 p5 B" }& D/ s! \
$ l9 D0 k1 |7 l# U' L6 @5 h* Mb=-kc+s
$ o9 e$ Q& v0 b& W
8 @# a. j6 A8 ]finally:  c= (s-b)/k

" j4 C2 S# m) h7 D* K9 Vthe right side cannnot contain c, but b contains c, so it is still not solved.

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-3 20:41:
4 {, i- X* |0 ]+ S9 w2 {请教大家一道微分题，多谢！
9 {! _; G. {, c  u; H+ e
d [(a+bx) ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
- j: Z) D3 Z, a, t
多谢了！

[ Last edited by 醉酒 ...

sorry, did you post another question? your statement is still not clear.
% g0 l* f4 G, e, N- U- Y
4 j) y4 v' O$ Xa, b, s, k are  constants? c means c(x) ?

4 `. |& }$ \# R8 t8 Kplease tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
sorry, did you post another question? your statement is still not clear.
' i4 ?- W6 y7 C! J5 H8 Z! o( K
a, b, s, k are  constants? c means c(x) ?
2 R. z( T9 j7 v' H0 d
( B3 Q6 Q  a! v- r1 \please tell me.

对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s
/ N& F  R- H) F/ B& O3 X
. X4 U" M6 J+ F) v/ f# F多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
, ~- T# j5 d9 b5 p7 i& x; l, m对不起，写错了，忘了变量c,应该是
7 V+ Z! B' K. I1 ~& \8 S4 f4 J7 b- A* }d [(a+bx) c] /dx = -kc +s
# x; K  Z6 P7 ^& r: i
多谢

- J" m& k/ j9 E3 n4 n: mdo you mean it is:
. I7 V; R) O+ \% i
( V6 r0 S- }+ t/ m+ Z" Id { (a+bx) C(x)}/dx = -k C(x) + s
1 F  k, [' D& _3 l3 \6 ]
, V2 x3 O' @2 I5 iwhere a, b, s are constants, you want to know what is C(x) ?

if so, this needs to solve the differential equation. please comfirm it first.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-4 06:36 PM:
do you mean it is:
6 W- O" V4 z6 ?2 V  \
d { (a+bx) C(x)}/dx = -k C(x) + s

* c  t  ]  d9 ewhere a, b, s are constants, you want to know what is C(x) ?

if so, this needs to solve the differential equation. please comfirm it fi ...

多谢你,你说对了,就是说用x的函数表示c. 也就是 c(x), 多谢你了。请帮忙。

十年移民路_

answer:

8 s7 X) w2 z. k7 z$ NC(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.
" b# Z0 z3 k1 t% \# f

procedure:
$ y+ O1 h- p7 b' \% t# ?5 C4 a; U
From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
8 G3 \# |* y, v* L/ m+ V: c9 Kso:
7 |7 V# z0 |! Q5 B+ U0 r( U
% c6 Y- v* n* k7 G, AbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.

(a+bx) dC(x)/dx  = -(k+b)C(x) +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
: ~% ~# p7 G8 _therefore:

/ W) l! n8 ^9 k2 r* P{(a+bx)/K} dY(x)/dx=Y(x)
8 M' [2 P- T0 `6 }- O6 V
from here, we can get:
+ }# \' n; q2 Q& S
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)

so that:   ln Y(x) =( K/b) ln(a+bx)
9 R) i  ]$ M7 O2 K' H+ B
7 `( Y& m( P' ^: v2 Cthis means:  Y(x) = (a+bx)^(K/b)
+ c/ i3 K" B- P; ~by using early transform, we can have:

-(k+b)C(x)+s = (a+bx)^(k/b+1)

finally:

0 a8 P3 y; x6 A% e) N2 FC(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)

醉酒当歌

果然是高手，多谢了，打印出来好好研究一下。高等数学学得实在不好。你一定是工程师吧，而且数学相当好。再谢！

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-5 14:33:
果然是高手，多谢了，打印出来好好研究一下。高等数学学得实在不好。你一定是工程师吧，而且数学相当好。再谢！

请你再仔细检查一遍小地方，但我可以保证方法和过程是没有问题的。
6 `2 e, F) e; C. T1 a6 ^是的，我现在是注册工程师，在上大学时，曾把“基米多维奇”和“樊映川”两本习题集通做一遍。（共大约一万多题吧）。这种题真的不是难题。

龙舞九天

9楼的答案有点问题吧。
, L! k% q. d! F8 r! {您说：
2 f# z  m. F; B
d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
& c+ a& ~% J6 k
也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
# X* t4 s# V* N) J但这是不正确的
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
0 R. Z8 j3 S. Z3 v$ j- }/ [7 ?& `并不是   (a+bx)*C(x)   的导数除以   x   的导数。
比如说   dx/dy = (x'y-xy')/y2
x' 意思是x的导数，y'意思是y的导数。分母是y的平方。
8 x4 l1 W8 v; l7 G2 g7 \+ h因此：
2 ^/ y2 @( t1 R" h
/ h) o% v. l' _, }d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
最右边是x的平方。

) H& }  R7 i' b; i所以您解的有问题。
- J! h- i( G3 {
[ Last edited by 龙舞九天 on 2005-4-6 at 07:35 PM ]

十年移民路_

Originally posted by 龙舞九天 at 2005-4-6 19:34:
! E% V# d3 P0 Q# Z9楼的答案有点问题吧。
6 n6 y# J9 n0 W$ V0 Z2 ?' m) W您说：

d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s

也就是说您认为   d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx
但这是不正确的
. c4 H# v) i0 P. ^7 hd{(a+bx)* ...

+ @0 U0 L: z' ]9 D  X
2 u+ J- w. p& T, H( }Please note here:
4 \6 c! I& P* V) w" N& t! B% }
d{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
3 K) H5 q8 i0 mit is a multiple, not division. Right ?

十年移民路_

但这是不正确的
d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
; R" t6 c5 _% y! [% ]' h并不是   (a+bx)*C(x)   的导数除以   x   的导数。
- g' ]1 {) U0 I0 p# |! M3 Q4 d
3 U- ^$ `% [, z* V' b7 T# u8 xno "x" in that position.

gnome-kde

d{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
  N; s" y" I( f) i3 C7 C7 q$ L....
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

这样算混淆了微分和导数的概念。

d(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算
8 m6 Q9 p2 U  X
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h'(x) = [xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2 (应该对吧， 我也没验算)
9 u1 J/ O7 G: ?9 i9 \
[ Last edited by gnome-kde on 2005-4-6 at 08:31 PM ]

十年移民路_

Originally posted by gnome-kde at 2005-4-6 20:30:
6 e( o6 D4 g, }4 G; L  [' K! [这样算混淆了微分和导数的概念。
8 `2 T0 S1 Q  ^, e! ?
  v8 @4 }" r7 T/ U& B& v4 c6 W% [1 Pd(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算
2 Y5 j- V# x4 W. E0 G. e6 s
If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

- m' X5 J  t8 }+ G这样算混淆了微分和导数的概念。
' `% d6 Z( i( L0 ^4 O5 c9 G2 |
% N% w/ ~+ p3 C" `  t4 b- h0 R) ~8 E* [d(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算
2 G; A5 h7 h5 G0 Y5 O# R' Q% ?* f
Ah, you made a mistake: it should be:
$ z: z7 J1 G0 e- _& B! `6 m, y
' z9 x6 V6 j! ?8 M0 @( md(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
+ O/ h! K1 E4 v7 |) V* }5 E

) ^( m5 F$ f$ c9 }3 zIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h'(x) = [xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2 (应该对吧， 我也没验算)

' Z+ E+ f# O# }& f7 Rfor h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:
, Y. c4 N/ q( i" a8 R
h(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
' i8 E, c8 W+ A, A这样算混淆了微分和导数的概念。

1 s) D  l! J' a6 L2 E- y  x) od(f(x))/dx = f'(x), d(f(x)g(x))/dx = f'(x) d(g(x))/dx + g'(x) d(f(x))/dx 这是基本的微分四则运算
" \/ r) B/ e9 h, H
) c5 l' o/ n9 B4 T2 TAh, you made a mistake: it should be:

1 g# j) y. |7 ]  ~5 td(f(x)g(x)) ...

$ l5 ?% t! N( \
. h: y2 @3 s) _  y# Y9 F; {Yes, there should not be " ' " after "f" and "g", it is a typo. :D
% W- J- w, f( F3 u
3 l) g4 g% I7 WBut for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

十年移民路_

Originally posted by gnome-kde at 2005-4-6 20:51:
Yes, there should not be " ' " after "f" and "g", it is a typo. :D
" e2 z. d) {' \) v# t) y  ~; }
$ D6 T& q9 M" j) pBut for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

- C. Q) o$ x! _0 ^
$ {9 U( `! k+ r7 [2 IThanks, I am happy here to discuss math.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-5 08:41 PM:

4 \& G" a4 Y. V; Z请你再仔细检查一遍小地方，但我可以保证方法和过程是没有问题的。
是的，我现在是注册工程师，在上大学时，曾把“基米多维奇”和“樊映川”两本习题集通做一遍。（共大约一万多题吧）。这种题真的 ...

( b+ ?5 d8 E0 s8 q  ~' ^" s: A佩服！是不是微分方程如果不能transform就不能求解了，有没有现成的微分方程求解公式？

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-6 23:24:
/ [& ~9 i  i: k佩服！是不是微分方程如果不能transform就不能求解了，有没有现成的微分方程求解公式？

理论上，只有很少的几类”齐次常微分方程“可以写出其”封闭的解析解“。不作变换，微分方程是非齐次的，可能不太容易作到。一般地说，没有什么现成的微分方程求解公式。但对”线性定常齐次微分方程“是有一个较方便的解法。