数学高手请进，多谢！

醉酒当歌 · 发表于 2005-4-3 19:41

请教大家一道微分题，多谢！

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?

多谢了！

[ Last edited by 醉酒当歌 on 2005-4-3 at 07:56 PM ]

十年移民路_ · 发表于 2005-4-3 19:52

Originally posted by 醉酒当歌 at 2005-4-3 20:41:  v! n" l) P( {; Y9 x6 }( e
请教大家一道微分题，多谢！
  D7 c/ J0 ~6 R/ |. n, n7 j  U+ x2 S4 F+ Z
d [(a+bx) ] /dx = -kc +s 7 ^  C0 |+ P  N3 e# `# i
where: only x and c are unknown, others are all known,  requiire c = ?2 E1 v1 o) S& {# x

7 x# l. Y, i. L: B多谢了！

it is too easy:

d(a+bx)/dx = b

so

b=-kc+s

finally: c= (s-b)/k

醉酒当歌 · 发表于 2005-4-3 19:57

Originally posted by 十年移民路 at 2005-4-3 07:52 PM:
. H+ r; @/ K) ~# `3 N8 ?it is too easy:) X/ R8 p" e7 X
$ j1 a" g' @, e. `  Y3 J
d(a+bx)/dx = b
9 ~# X/ [& `) ^8 D/ P
6 y; L' t# {6 e! D9 x0 }" D) T  eso
0 X4 t. Z% _. C3 x- ~# u2 _
3 P5 |8 M2 H. t# G/ pb=-kc+s' q) ~7 Y9 o  @0 p) J
7 X4 B7 L( R; }8 o/ d
finally:  c= (s-b)/k

the right side cannnot contain c, but b contains c, so it is still not solved.

十年移民路_ · 发表于 2005-4-3 20:51

Originally posted by 醉酒当歌 at 2005-4-3 20:41:5 [, I, D0 m/ D
请教大家一道微分题，多谢！9 X, o+ {$ Y) @% ?

- Z; N& U! p. [7 }2 _4 _d [(a+bx) ] /dx = -kc +s / p: Z$ u0 }: ^
where: only x and c are unknown, others are all known, requiire c = function of x, what is this function?
$ y' _/ b( ?* P0 ^9 ]- |* [9 z
. ]% g6 C3 I3 W# `+ J多谢了！6 g3 v0 w4 x9 p% w& t. Y
0 ~1 b4 F% ~1 V. H3 ?6 f" v* G
[ Last edited by 醉酒 ...

sorry, did you post another question? your statement is still not clear.

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

please tell me.

醉酒当歌 · 发表于 2005-4-4 11:30

Originally posted by 十年移民路 at 2005-4-3 08:51 PM:, f6 b0 r( v3 y
sorry, did you post another question? your statement is still not clear.
9 o, x5 A: n( Z3 }* I
, n2 u, r- J! k. aa, b, s, k are constants? c means c(x) ?
8 n( @! L; W% M% l% b. L, \2 O" W! a0 u6 m+ N
please tell me.

对不起，写错了，忘了变量c,应该是
d [(a+bx) c] /dx = -kc +s

多谢

十年移民路_ · 发表于 2005-4-4 18:36

Originally posted by 醉酒当歌 at 2005-4-4 12:30:8 @6 M. p$ U" w( X: A
对不起，写错了，忘了变量c,应该是9 g+ Z% l" F3 T) W: G
d [(a+bx) c] /dx = -kc +s
5 ]% a9 k! p {6 W/ w3 J, Q: z3 ~8 @/ D
多谢

do you mean it is:

d { (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 · 发表于 2005-4-4 18:43

Originally posted by 十年移民路 at 2005-4-4 06:36 PM:8 @. q7 `2 ~- r9 C8 G D
do you mean it is:- \* o% M* n, o% D' U: D5 K: R6 x
$ d% E" K5 `# O# U# U$ u- _7 s
d { (a+bx) C(x)}/dx = -k C(x) + s7 r' }" }) \; _8 A9 G4 r( Z) u
$ h3 J$ J, W8 j
where a, b, s are constants, you want to know what is C(x) ?
1 w" P* p5 }2 D0 ?- \6 x7 C" D, s1 _* I9 T0 k8 H
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_ · 发表于 2005-4-4 19:17

answer:

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

十年移民路_ · 发表于 2005-4-4 22:26

this answer is the good one.

procedure:

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

bC(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
therefore:

{(a+bx)/K} dY(x)/dx=Y(x)

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)

so that: ln Y(x) =( K/b) ln(a+bx)

this means:  Y(x) = (a+bx)^(K/b)
by using early transform, we can have:

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

finally:

C(x)= -1/(k+b)*{[(a+bx)^(k/b+1)] - s)

醉酒当歌 · 发表于 2005-4-5 13:33

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！

十年移民路_ · 发表于 2005-4-5 20:41

Originally posted by 醉酒当歌 at 2005-4-5 14:33:
: P+ r n3 L! Y7 l L& B% [2 S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。
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 · 发表于 2005-4-6 18:34

9楼的答案有点问题吧。
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
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。
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
最右边是x的平方。

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_ · 发表于 2005-4-6 18:53

Originally posted by 龙舞九天 at 2005-4-6 19:34:
  }: s9 S( w- t. O6 I" Q9楼的答案有点问题吧。- `; G/ u9 `" p6 B: x$ K
您说：4 ^7 }, M4 J: b$ r$ J
' a- L: O; n' g; \
d{(a+bx)*C(x)}/dx =-k C(x) + s% y! ~5 K4 l. R% ~; y4 ^5 h; g
so:1 ~  z) f5 g6 R9 w, ]
bC(x) + (a+bx) dC(x)/dx  = -kC(x) +s. u0 S0 _& z0 x5 R1 Y
% q# T( P' \+ O& k# [$ v
也就是说您认为 d{(a+bx)*C(x)}/dx = bC(x) + (a+bx) dC(x)/dx/ Z( x1 E" g) ?4 ^  V* J+ V1 E
但这是不正确的
# R% E0 _4 N3 w# Md{(a+bx)* ...

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_ · 发表于 2005-4-6 18:56

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。

no "x" in that position.

gnome-kde · 发表于 2005-4-6 19:30

d{(a+bx)*C(x)}/dx 意思是说对 (a+bx)*C(x)/x 整个代数式求导数% e# {# z! k: u" C# j3 N
....
+ N/ V0 W! ?! I3 F8 P5 z4 ed{(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

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)

[ Last edited by gnome-kde on 2005-4-6 at 08:31 PM ]

十年移民路_ · 发表于 2005-4-6 19:45

Originally posted by gnome-kde at 2005-4-6 20:30:
( P$ b5 i( S' P+ I这样算混淆了微分和导数的概念。7 ?4 {# t" }! q
" M$ o# O F: |5 V+ p) T
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
) P1 l; L3 ?' n% x" m0 Z
* l/ _; ?% i8 F3 n* K& WIf 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:

d(f(x)g(x))/dx = f(x) d(g(x))/dx + g(x) d(f(x))/dx

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)

for h(x)=(a+bx)*C(x)/x, you are right, But, no x there, it is:

h(x)=(a+bx)*C(x)

gnome-kde · 发表于 2005-4-6 19:51

Originally posted by 十年移民路 at 2005-4-6 08:45 PM:+ t1 p. p( h6 [$ s1 A6 f1 X
这样算混淆了微分和导数的概念。' j6 K  q3 t, x) X' t: a: H% ?

- l5 a, i  J: \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 `3 b8 Z( ]7 U  R/ a4 ?. ]0 Y8 }0 m5 L" \( b+ U; d5 d
Ah, you made a mistake: it should be:% U& Q1 g; M+ G- F- L

/ |( J! z/ v* Y: Y, ^5 f* f, Ed(f(x)g(x)) ...

Yes, there should not be " ' " after "f" and "g", it is a typo. :D

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_ · 发表于 2005-4-6 20:05

Originally posted by gnome-kde at 2005-4-6 20:51:7 [3 f3 o: F7 z4 @# @0 b
Yes, there should not be " ' " after "f" and "g", it is a typo. :D
9 [6 m# X6 h2 o, m- a; T! A2 P1 O( ]
But 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 · 发表于 2005-4-6 22:24

Originally posted by 十年移民路 at 2005-4-5 08:41 PM:3 ~* s) @& B, v& p6 F9 @
( z6 T1 h* K1 n) q/ m
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。
# w+ O: G. w* c4 u5 h* Z' 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 ...

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_ · 发表于 2005-4-7 20:44

Originally posted by 醉酒当歌 at 2005-4-6 23:24:8 _! w) N J& N
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。

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