数学高手请进，多谢！

醉酒当歌

请教大家一道微分题，多谢！
& N* h6 \0 N) L
" E3 v9 X! a- N8 S. {! H( d( md [(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?
# }; J8 v" O% R4 c) `4 n
' L0 Z  F! {5 c1 V: P3 J8 X多谢了！
# P  I* \' e' T) Z. n. P1 Z4 c
[ 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！
; @1 V$ |/ j8 R
d [(a+bx) ] /dx = -kc +s
where: only x and c are unknown, others are all known,  requiire c = ?

多谢了！

0 [) x. w9 u$ x, ]5 \
it is too easy:
+ L& {" Y1 V, `9 ]1 Z% y! V+ `
) t$ c- t$ Y; a  N8 g8 ld(a+bx)/dx = b
& q$ O0 Y% h9 ^0 O  l
8 Z; U. ?( S; u9 S! tso

2 t( n7 L! _" d) b+ cb=-kc+s
! d/ z) n7 w0 F
finally:  c= (s-b)/k

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
$ n3 r0 S4 n( n  r$ ^' a% yit is too easy:
6 D% r9 s9 o" w4 v2 F. Q% o
3 p6 {. ?( n7 [/ p% ~4 id(a+bx)/dx = b

so
7 |# O2 y- u, Y) W; L
b=-kc+s

finally:  c= (s-b)/k

/ W) b. J, Z) B
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！
9 |- a, u& C: I- I
$ u1 C9 [  C( f/ e& M3 R( q) @d [(a+bx) ] /dx = -kc +s
$ y0 P! r6 M; |8 R0 ]where: only x and c are unknown, others are all known,  requiire c = function of x, what is this function?
: w! {+ p. e3 e; N: ]
$ o+ G( g* D; n' H% v多谢了！
/ m! u2 Z* j; E1 `+ w3 i
[ Last edited by 醉酒 ...

+ Y! }2 g6 r# u7 x, u( U* K) Hsorry, did you post another question? your statement is still not clear.
% z0 y. X4 ^- v: x% i/ D7 L- o
a, b, s, k are  constants? c means c(x) ?
; b5 d0 v- |, h
please tell me.

醉酒当歌

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:
8 X, A0 R4 d* j  T3 }! t! fsorry, did you post another question? your statement is still not clear.
2 I3 J1 `) O2 h& \: q6 [+ n$ \; n
a, b, s, k are  constants? c means c(x) ?
; p8 ^# N* T* R+ A( H; a" b& l4 \
- A( ^' p. V3 \+ s' aplease tell me.

对不起，写错了，忘了变量c,应该是
/ L5 q) _3 P) P0 A. x( j5 k) Cd [(a+bx) c] /dx = -kc +s
9 Y9 M- L* j; w' M2 i
多谢

十年移民路_

Originally posted by 醉酒当歌 at 2005-4-4 12:30:
, ]9 W# I( x3 R0 \6 n. l对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s
8 Q( y! C, x/ _  f6 x& N  }
多谢

do you mean it is:
  z" w2 H: d$ g, T  D
& y1 W) c6 S( U/ s; ^& T5 Nd { (a+bx) C(x)}/dx = -k C(x) + s

. h; G1 _$ o; ?$ Xwhere 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:
, A& P7 X7 M- m! _/ {# b2 v0 Ido you mean it is:
* \4 O! L5 y' U. ~
d { (a+bx) C(x)}/dx = -k C(x) + s
9 ]0 P' a3 `+ u' O/ o
4 D* f+ P8 Q8 r$ \  Y& d1 b1 Fwhere a, b, s are constants, you want to know what is C(x) ?

8 `; ^  c) {' D( H6 p9 iif so, this needs to solve the differential equation. please comfirm it fi ...

5 i, `' u, V4 E( [  r' i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:
9 Y$ V- f) r+ H1 b2 E( [
C(x)=-(1/k) * {(a+bx)^(-k/b) } - (s/k)

十年移民路_

this answer is the good one.
& |7 T" U0 r) w/ d. P( N
4 s- T' ~# U- D% \
% M/ `. U( j! u- Bprocedure:

From:  d{(a+bx)*C(x)}/dx =-k C(x) + s
9 U) p' P5 }. K# @* U5 l1 Vso:
/ n% C: W, Y( C8 z7 A% Q! i5 M
+ B4 x' a2 K$ RbC(x) + (a+bx) dC(x)/dx = -kC(x) +s
i.e.

(a+bx) dC(x)/dx  = -(k+b)C(x) +s
( Q$ T2 n# @1 Y* q8 d( A$ g

3 C+ }/ ^% F% c) a6 g, N: Aintroduce a tranform: KC(x)+s =Y(x), where K=-(k+b)
5 V7 Z* Q6 Z! Y9 L4 u# J0 _which means: KdC(x)/dx = dY(x)/dx or dC(x)/dx = (1/K)dY/dx
) a- P4 A6 u1 t# A5 c" Q) F' @therefore:
$ ~, n4 U6 {+ q: x! H, }
' L+ V; h3 K7 u6 o1 D$ O{(a+bx)/K} dY(x)/dx=Y(x)
2 I! F3 n# w" T$ I$ Z1 \' @8 R
: {4 s/ e& I; ^6 K# p/ N' u0 u5 H& Cfrom here, we can get:
5 @, x7 S, z3 k8 n9 a; H. ^
dY(x)/Y(x) = [K/(a+bx)]dx  i.e. dY(x)/Y(x) = {K/b(a+bx)}d(a+bx)
% e  F4 Y% u; ]5 l# B. V
" }/ [( X- @' ~4 v+ G* Bso that:   ln Y(x) =( K/b) ln(a+bx)
6 d( l) Z4 n1 L: l
( @; b* k# O* p0 j$ t% Z" othis means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:
% w, G$ q8 @" J% \1 h. v) i
-(k+b)C(x)+s = (a+bx)^(k/b+1)

% k, \3 |* {8 S+ d6 vfinally:

  \+ U% T  t+ d' Z9 ~% W8 _C(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:
: w% k$ N2 \0 M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！

. [9 {9 L! u: ]
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。
4 A& q5 N+ }- f9 a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：
# _" o5 T( L4 B( U+ `% I/ {3 w- E
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
9 a' ~6 S! i  C+ `" t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
1 P$ s1 M" {/ x7 H) b$ O+ u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。
因此：

d{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2
5 Z4 H* e/ P1 |, J0 Q% M! {3 K最右边是x的平方。
6 T. I. v( H1 {! S2 {3 I
所以您解的有问题。
* ^7 |' p! H0 ^% _
2 P6 a& u5 ?8 I[ 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:
2 i+ s7 h$ b* X9楼的答案有点问题吧。
您说：

8 O6 D$ M( j6 Y4 g& Vd{(a+bx)*C(x)}/dx =-k C(x) + s
so:
* W- H# j6 z6 abC(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
# h" M' v: s/ L5 H+ e' K6 F' ?但这是不正确的
d{(a+bx)* ...

# y; G5 |( ~- O0 {Please note here:

d{(a+bx)*C(x)}/dx  means:   df(x)/dx, where f(x) = (a+bx)*C(x)
it is a multiple, not division. Right ?

十年移民路_

但这是不正确的
1 Y4 L& L, H1 Bd{(a+bx)*C(x)}/dx  意思是说对   (a+bx)*C(x)/x   整个代数式求导数
9 ~% H5 @& V1 z4 C3 B) |7 u# @并不是   (a+bx)*C(x)   的导数除以   x   的导数。

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 }& q4 ?$ D& t+ ]1 Md{(a+bx)*C(x)}/dx =[xb*c(x)+x(a+bx)*c'(x)-(a+bx)*c(x)]/x2

# K" ~3 {- D7 i: Y$ g) b6 C4 ?9 W" }
这样算混淆了微分和导数的概念。
% A+ b% S! Q( D$ H
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
3 x; {, Q( A4 g: T3 g1 V0 h& c
! i- f2 Y$ F6 I: F4 TIf 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)
% y5 R+ B( _6 u( i( }# m$ _. {
7 T" `! e6 a6 p( F4 y$ `[ 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。
0 v3 \: d& H/ _: z1 \; e
5 p7 P9 \6 w% ]! D* B! P5 {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

If h(x)=(a+bx)*C(x)/x, then d(h(x))/dx = d{(a+bx)*C(x)/x}/dx = h ...

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

Ah, you made a mistake: it should be:

4 {$ D% o1 V2 k" S5 Wd(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx
+ y# V8 `4 T( B! [: @4 B. M- X2 ]

" k1 t6 D# c  G- v2 w( Z1 mIf 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)
$ s) s$ f9 Q' i* f; m$ c& j
0 i; t. J, f! N) L: V' E9 {for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

# r. w. I: K8 K% T* [2 c* g( Zh(x)=(a+bx)*C(x)

gnome-kde

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:
3 u) \0 L8 s. y# W, O, J) ]6 D这样算混淆了微分和导数的概念。
4 b( g4 m$ y! Y+ A0 V
1 F* A( |+ }6 p5 X# sd(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 i1 X5 M0 ^9 k( f4 |
/ g( o9 p5 |9 u  o# f# zAh, you made a mistake: it should be:
+ i/ W) ~  Y: n# x
+ e, e  H( ~1 Vd(f(x)g(x)) ...

5 ^3 e2 v8 A9 _) N
( f! _& u0 b4 A+ y( I1 XYes, there should not be " ' " after "f" and "g", it is a typo. :D

6 w% z& g0 Q0 X( ~: hBut 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:
9 H0 Z2 U) i) l- y3 zYes, there should not be " ' " after "f" and "g", it is a typo. :D

9 ?3 a% P. H% i2 J0 P: z. a7 Z3 ABut for the left part, I am not intented to show you are wrong but "龙舞九天" is wrong.

Thanks, 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:
  U- u) h1 w& I0 u" [- F
" S9 N, B6 B/ K0 b3 y1 S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。
# R% w  G8 E. }3 J. u) Y# y( Y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 ...

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:
0 {3 N& Y8 d* I5 o9 U( 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？

9 I3 D, ]& H/ Y
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。