数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！
8 `: f) N, E( J+ }9 F! f9 J  s
. @" N2 _6 B7 K" k* Id [(a+bx) ] /dx = -kc +s
5 \: @. o8 p, ?where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
& b# [9 W/ J% K+ }! t' p
多谢了！

1 g, z& U/ \; S  G9 l* h9 S9 U[ 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！
! {6 b! x. P. T/ e- t" D! N
d [(a+bx) ] /dx = -kc +s
* k3 K; }7 r+ Y8 t& F) qwhere: only x and c are unknown, others are all known,  requiire c = ?
7 c( X5 A& D- n/ t& |+ Q
. L+ g' K1 T( i& f5 N; |多谢了！

7 r. e, U" |  r# ^* J" z
; C; f, Q; S8 a, _1 eit is too easy:

( d: @+ S% _) b8 }' j: ?d(a+bx)/dx = b

so

b=-kc+s

, b# [2 s$ P& efinally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
- C1 A7 X& z8 \% _+ }1 ^" N5 oit is too easy:
1 {" T2 o: t  O& K* \9 d! q* L0 w
d(a+bx)/dx = b

so

! o1 l% n! }* s2 Q/ p8 p  Vb=-kc+s
% ]6 f% {, d, E8 ~# b
: B9 _  E. a, Z+ N# I4 _: wfinally:  c= (s-b)/k

+ q" k* P& I  m6 Y# Q7 i7 _
4 M/ }( E% S+ S- {& c9 ?1 @the 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:
请教大家一道微分题，多谢！
8 i* f7 ]( t. \6 Q( e9 D
d [(a+bx) ] /dx = -kc +s
/ e8 n! [" i/ c3 ^( h# twhere: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
! h8 N) j* A$ Z  m: i! Q
多谢了！
- Z0 x1 R! b  W+ p! J- ~
[ Last edited by 醉酒 ...

% D- B$ }! D0 E+ qsorry, did you post another question? your statement is still not clear.

a, b, s, k are  constants? c means c(x) ?
1 v! o. k& T; a$ R
  M& Q5 M  Y( dplease tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
' Z  v% o  f! h: ~8 J+ J' P. csorry, did you post another question? your statement is still not clear.

a, b, s, k are  constants? c means c(x) ?

please tell me.

1 }  m2 h: x3 }9 H对不起，写错了，忘了变量c,应该是
4 K& p- A* p6 y8 d2 q5 {, b6 \d [(a+bx) c] /dx = -kc +s

" [# D: {+ T2 W/ \: [; @3 E; I4 e多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s

多谢

' T9 R9 o% r* }+ H9 @4 t2 ado you mean it is:
/ b3 c  u  X: y9 w  r
9 `) [! @" ^5 \  n8 J, T2 Bd { (a+bx) C(x)}/dx = -k C(x) + s

where 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:
; F" W, B) T- r" v/ l0 V7 A
  P0 {: k/ l0 x, H: ^! [9 v. \7 od { (a+bx) C(x)}/dx = -k C(x) + s

where a, b, s are constants, you want to know what is C(x) ?
0 }; n9 C- i: b( V& v, u+ Y
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:
- j: `: J% J5 A) T* d8 w
: b+ u8 u! W# I" _9 _- _2 pC(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.
( o4 Y! J: X3 z: y% H0 Z
) P# G! \& g4 ?6 G4 s/ J
, I3 Y2 n% r) G" l/ k" V9 K9 Wprocedure:

  B! t7 M; B" Q# ~From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
+ j  g2 @6 I1 }' r  ^. Q$ Lso:
) O( g* I/ U% j
- w* v2 D9 q  h, j4 u3 PbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.

( ]" \" [" X: l" D/ K8 D(a+bx) dC(x)/dx  = -(k+b)C(x) +s
6 p+ T5 F: p" M0 w$ L/ d4 B9 d. v
$ e: S# N' k' [- f& d3 ]
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
therefore:
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{(a+bx)/K} dY(x)/dx=Y(x)
- ^( L+ {3 S3 j- ]: ?
from here, we can get:

dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)
) x4 [6 Y- t1 I! G  V
so that:   ln Y(x) =( K/b) ln(a+bx)

$ C# J8 P" k* \+ N2 W+ Dthis means:  Y(x) = (a+bx)^(K/b)
: H3 Z2 O* u) C7 W2 b6 _# Wby using early transform, we can have:

3 @: d8 X& `( \0 v# F4 U# {-(k+b)C(x)+s = (a+bx)^(k/b+1)

finally:

: X$ W1 s& L  M" G8 S( U' u* JC(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！

) j& I* f. B, a' }0 @, ?* t
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）。这种题真的不是难题。

龙舞九天

9楼的答案有点问题吧。
您说：
9 \; Y  r0 a, W5 j) d
d{(a+bx)*C(x)}/dx =-k C(x) + s
so:
+ m' `) [% {( ]: \/ v3 NbC(x) + (a+bx) dC(x)/dx  = -kC(x) +s
, |6 s0 q0 o! W5 s" X1 E3 w7 Q$ g
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
9 ]6 s/ s2 `+ K/ t7 Y7 w' ld{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
( \/ x- d$ S1 Y6 M" j, l并不是   (a+bx)*C(x)   的导数除以   x   的导数。
* n" w( M9 v4 h- Z8 u+ J1 G比如说   dx/dy = (x'y-xy')/y2
. K: w& Q) C, k" H4 |5 Fx' 意思是x的导数，y'意思是y的导数。分母是y的平方。
因此：
8 D* i" Z  N! C9 Y) X6 J$ @" r
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。
% R# Y% z/ G& P' d- j+ U# G
7 [% G2 M2 k# V5 \7 D% B% C所以您解的有问题。

[ 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:
! A/ d  L' ~) K2 k1 R; |$ D9楼的答案有点问题吧。
7 L& f3 w! y; e, e! v# B7 P您说：
2 v* H8 r! P1 p2 ?* h) t) Q
  |0 v! A" c& w& q$ N7 S  }" sd{(a+bx)*C(x)}/dx =-k C(x) + s
so:
$ e# Y) L; S& f1 gbC(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
d{(a+bx)* ...

! e9 v' h, z5 t$ A) t+ K, s+ {) I2 a
Please note here:
1 i+ O% ~4 U) L8 W/ d8 ?7 {
8 F6 `1 V8 Y- U0 ]2 U1 ^; Id{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
; M7 \7 P8 x) C% z9 [  bit 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
并不是   (a+bx)*C(x)   的导数除以   x   的导数。
8 P! S) {4 B/ Z  c
no "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
3 W* q: Q, S: a( T....
d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

/ P/ F% N8 [" o8 b1 w5 ~
2 u! R; z8 m. O这样算混淆了微分和导数的概念。
" @% Z7 g8 R3 d( q" j, A) b$ ^
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
, j5 r2 Z' ^" W/ T$ I" R6 N
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)

4 r5 x( P- ?, p2 c( p$ B- a. ^[ 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:
这样算混淆了微分和导数的概念。
4 Y, ^% W+ R* `* a  q4 {
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

% B8 H' ]9 m# F. V3 GIf h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

/ d  n% y* _4 J) d8 F
这样算混淆了微分和导数的概念。
% w% d/ Y) F6 d& S0 C. N9 e/ c
0 y  y: N3 I  Wd(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
1 {9 c: J3 Y% d; U7 W2 c" a
. G( ]% T* S: z! T7 w8 `" p; v7 B; TAh, you made a mistake: it should be:
6 {1 A; g2 P! D- h4 ?  u5 a* s
d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
( M* I7 `4 K& L' A: @
: g/ |1 d( A0 ?6 C4 k+ Z1 I" r
7 }  F; U' K2 E6 `/ 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)

  U9 K, m; w/ ?( d, [$ C; |& Wfor h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:
0 `* N* r8 c+ N  E8 [$ f
h(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
$ |; H6 j7 X  |0 p" L- }( R这样算混淆了微分和导数的概念。
/ c4 w5 L# u1 A
7 ?: p! [2 L. {$ b/ r. [+ id(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

Ah, you made a mistake: it should be:
# `, ~3 d  M6 l: z: g; `0 M+ n
d(f(x)g(x)) ...

Yes, there should not be " ' " after "f" and "g", it is a typo. :D
6 k1 H. Y: }( F* x9 H
But 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
% ^7 d3 @" s- h* d2 `2 c
, O1 f7 G! n+ ?7 q+ JBut for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

( {+ D* ~& T, G2 H9 n, T
4 R5 }5 o: G5 o! vThanks, 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:
0 R6 n3 U& u5 m6 n% b3 i/ Y
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 ...

3 h+ x% t( t! S/ e6 ~! L) 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？

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-6 23:24:
$ K( S1 L" n  _9 w# J" A3 V9 B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？

/ i7 u% o% \! P& J
3 V, @. ^4 U* }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。